To create various kinds of inequalities, the idea of convexity is essential. Convexity and integral inequality hence have a significant link. This study's goals are to introduce a new class of generalized convex fuzzy-interval-valued functions (convex 𝘍𝘐𝘝𝘍s) which are known as (p,J)-convex 𝘍𝘐𝘝𝘍s and to establish Jensen, Schur and Hermite-Hadamard type inequalities for (p,J)-convex 𝘍𝘐𝘝𝘍s using fuzzy order relation. The Kulisch-Miranker order relation, which is based on interval space, is used to define this fuzzy order relation level-wise. Additionally, we have demonstrated that, as special examples, our conclusions encompass a sizable class of both new and well-known inequalities for (p,J)-convex 𝘍𝘐𝘝𝘍s. We offer helpful examples that demonstrate the theory created in this study's application. These findings and various methods might point the way in new directions for modeling, interval-valued functions and fuzzy optimization issues.
Citation: Muhammad Bilal Khan, Gustavo Santos-García, Hüseyin Budak, Savin Treanțǎ, Mohamed S. Soliman. Some new versions of Jensen, Schur and Hermite-Hadamard type inequalities for (p,J)-convex fuzzy-interval-valued functions[J]. AIMS Mathematics, 2023, 8(3): 7437-7470. doi: 10.3934/math.2023374
[1] | Muhammad Ghaffar Khan, Nak Eun Cho, Timilehin Gideon Shaba, Bakhtiar Ahmad, Wali Khan Mashwani . Coefficient functionals for a class of bounded turning functions related to modified sigmoid function. AIMS Mathematics, 2022, 7(2): 3133-3149. doi: 10.3934/math.2022173 |
[2] | Rabha W. Ibrahim, Dumitru Baleanu . Fractional operators on the bounded symmetric domains of the Bergman spaces. AIMS Mathematics, 2024, 9(2): 3810-3835. doi: 10.3934/math.2024188 |
[3] | Muhammmad Ghaffar Khan, Wali Khan Mashwani, Jong-Suk Ro, Bakhtiar Ahmad . Problems concerning sharp coefficient functionals of bounded turning functions. AIMS Mathematics, 2023, 8(11): 27396-27413. doi: 10.3934/math.20231402 |
[4] | Muhammmad Ghaffar Khan, Wali Khan Mashwani, Lei Shi, Serkan Araci, Bakhtiar Ahmad, Bilal Khan . Hankel inequalities for bounded turning functions in the domain of cosine Hyperbolic function. AIMS Mathematics, 2023, 8(9): 21993-22008. doi: 10.3934/math.20231121 |
[5] | Zhen Peng, Muhammad Arif, Muhammad Abbas, Nak Eun Cho, Reem K. Alhefthi . Sharp coefficient problems of functions with bounded turning subordinated to the domain of cosine hyperbolic function. AIMS Mathematics, 2024, 9(6): 15761-15781. doi: 10.3934/math.2024761 |
[6] | Lina Ma, Shuhai Li, Huo Tang . Geometric properties of harmonic functions associated with the symmetric conjecture points and exponential function. AIMS Mathematics, 2020, 5(6): 6800-6816. doi: 10.3934/math.2020437 |
[7] | Xinghua You, Ghulam Farid, Lakshmi Narayan Mishra, Kahkashan Mahreen, Saleem Ullah . Derivation of bounds of integral operators via convex functions. AIMS Mathematics, 2020, 5(5): 4781-4792. doi: 10.3934/math.2020306 |
[8] | İbrahim Aktaş . On some geometric properties and Hardy class of q-Bessel functions. AIMS Mathematics, 2020, 5(4): 3156-3168. doi: 10.3934/math.2020203 |
[9] | Muhammad Ghaffar Khan, Sheza.M. El-Deeb, Daniel Breaz, Wali Khan Mashwani, Bakhtiar Ahmad . Sufficiency criteria for a class of convex functions connected with tangent function. AIMS Mathematics, 2024, 9(7): 18608-18624. doi: 10.3934/math.2024906 |
[10] | Yue Wang, Ghulam Farid, Babar Khan Bangash, Weiwei Wang . Generalized inequalities for integral operators via several kinds of convex functions. AIMS Mathematics, 2020, 5(5): 4624-4643. doi: 10.3934/math.2020297 |
To create various kinds of inequalities, the idea of convexity is essential. Convexity and integral inequality hence have a significant link. This study's goals are to introduce a new class of generalized convex fuzzy-interval-valued functions (convex 𝘍𝘐𝘝𝘍s) which are known as (p,J)-convex 𝘍𝘐𝘝𝘍s and to establish Jensen, Schur and Hermite-Hadamard type inequalities for (p,J)-convex 𝘍𝘐𝘝𝘍s using fuzzy order relation. The Kulisch-Miranker order relation, which is based on interval space, is used to define this fuzzy order relation level-wise. Additionally, we have demonstrated that, as special examples, our conclusions encompass a sizable class of both new and well-known inequalities for (p,J)-convex 𝘍𝘐𝘝𝘍s. We offer helpful examples that demonstrate the theory created in this study's application. These findings and various methods might point the way in new directions for modeling, interval-valued functions and fuzzy optimization issues.
Let A denote the class of functions f which are analytic in the open unit disk Δ={z∈C:|z|<1}, normalized by the conditions f(0)=f′(0)−1=0. So each f∈A has series representation of the form
f(z)=z+∞∑n=2anzn. | (1.1) |
For two analytic functions f and g, f is said to be subordinated to g (written as f≺g) if there exists an analytic function ω with ω(0)=0 and |ω(z)|<1 for z∈Δ such that f(z)=(g∘ω)(z).
A function f∈A is said to be in the class S if f is univalent in Δ. A function f∈S is in class C of normalized convex functions if f(Δ) is a convex domain. For 0≤α≤1, Mocanu [23] introduced the class Mα of functions f∈A such that f(z)f′(z)z≠0 for all z∈Δ and
ℜ((1−α)zf′(z)f(z)+α(zf′(z))′f′(z))>0(z∈Δ). | (1.2) |
Geometrically, f∈Mα maps the circle centred at origin onto α-convex arcs which leads to the condition (1.2). The class Mα was studied extensively by several researchers, see [1,10,11,12,24,25,26,27] and the references cited therein.
A function f∈S is uniformly starlike if f maps every circular arc Γ contained in Δ with center at ζ ∈Δ onto a starlike arc with respect to f(ζ). A function f∈C is uniformly convex if f maps every circular arc Γ contained in Δ with center ζ ∈Δ onto a convex arc. We denote the classes of uniformly starlike and uniformly convex functions by UST and UCV, respectively. For recent study on these function classes, one can refer to [7,9,13,19,20,31].
In 1999, Kanas and Wisniowska [15] introduced the class k-UCV (k≥0) of k-uniformly convex functions. A function f∈A is said to be in the class k-UCV if it satisfies the condition
ℜ(1+zf″(z)f′(z))>k|zf′(z)f′(z)|(z∈Δ). | (1.3) |
In recent years, many researchers investigated interesting properties of this class and its generalizations. For more details, see [2,3,4,14,15,16,17,18,30,32,35] and references cited therein.
In 2015, Sokół and Nunokawa [33] introduced the class MN, a function f∈MN if it satisfies the condition
ℜ(1+zf″(z)f′(z))>|zf′(z)f(z)−1|(z∈Δ). |
In [28], it is proved that if ℜ(f′)>0 in Δ, then f is univalent in Δ. In 1972, MacGregor [21] studied the class B of functions with bounded turning, a function f∈B if it satisfies the condition ℜ(f′)>0 for z∈Δ. A natural generalization of the class B is B(δ1) (0≤δ1<1), a function f∈B(δ1) if it satisfies the condition
ℜ(f′(z))>δ1(z∈Δ;0≤δ1<1), | (1.4) |
for details associated with the class B(δ1) (see [5,6,34]).
Motivated essentially by the above work, we now introduce the following class k-Q(α) of analytic functions.
Definition 1. Let k≥0 and 0≤α≤1. A function f∈A is said to be in the class k-Q(α) if it satisfies the condition
ℜ((zf′(z))′f′(z))>k|(1−α)f′(z)+α(zf′(z))′f′(z)−1|(z∈Δ). | (1.5) |
It is worth mentioning that, for special values of parameters, one can obtain a number of well-known function classes, some of them are listed below:
1. k-Q(1)=k-UCV;
2. 0-Q(α)=C.
In what follows, we give an example for the class k-Q(α).
Example 1. The function f(z)=z1−Az(A≠0) is in the class k-Q(α) with
k≤1−b2b√b(1+α)[b(1+α)+2]+4(b=|A|). | (1.6) |
The main purpose of this paper is to establish several interesting relationships between k-Q(α) and the class B(δ) of functions with bounded turning.
To prove our main results, we need the following lemmas.
Lemma 1. ([8]) Let h be analytic in Δ with h(0)=1, β>0 and 0≤γ1<1. If
h(z)+βzh′(z)h(z)≺1+(1−2γ1)z1−z, |
then
h(z)≺1+(1−2δ)z1−z, |
where
δ=(2γ1−β)+√(2γ1−β)2+8β4. | (2.1) |
Lemma 2. Let h be analytic in Δ and of the form
h(z)=1+∞∑n=mbnzn(bm≠0) |
with h(z)≠0 in Δ. If there exists a point z0(|z0|<1) such that |argh(z)|<πρ2(|z|<|z0|) and |argh(z0)|=πρ2 for some ρ>0, then z0h′(z0)h(z0)=iℓρ, where
ℓ:{ℓ≥n2(c+1c)(argh(z0)=πρ2),ℓ≤−n2(c+1c)(argh(z0)=−πρ2), |
and (h(z0))1/ρ=±ic(c>0).
This result is a generalization of the Nunokawa's lemma [29].
Lemma 3. ([37]) Let ε be a positive measure on [0,1]. Let ϝ be a complex-valued function defined on Δ×[0,1] such that ϝ(.,t) is analytic in Δ for each t∈[0,1] and ϝ(z,.) is ε-integrable on [0,1] for all z∈Δ. In addition, suppose that ℜ(ϝ(z,t))>0, ϝ(−r,t) is real and ℜ(1/ϝ(z,t))≥1/ϝ(−r,t) for |z|≤r<1 and t∈[0,1]. If ϝ(z)=∫10ϝ(z,t)dε(t), then ℜ(1/ϝ(z))≥1/ϝ(−r).
Lemma 4. ([22]) If −1≤D<C≤1, λ1>0 and ℜ(γ2)≥−λ1(1−C)/(1−D), then the differential equation
s(z)+zs′(z)λ1s(z)+γ2=1+Cz1+Dz(z∈Δ) |
has a univalent solution in Δ given by
s(z)={zλ1+γ2(1+Dz)λ1(C−D)/Dλ1∫z0tλ1+γ2−1(1+Dt)λ1(C−D)/Ddt−γ2λ1(D≠0),zλ1+γ2eλ1Czλ1∫z0tλ1+γ2−1eλ1Ctdt−γ2λ1(D=0). |
If r(z)=1+c1z+c2z2+⋯ satisfies the condition
r(z)+zr′(z)λ1r(z)+γ2≺1+Cz1+Dz(z∈Δ), |
then
r(z)≺s(z)≺1+Cz1+Dz, |
and s(z) is the best dominant.
Lemma 5. ([36,Chapter 14]) Let w, x and\ y≠0,−1,−2,… be complex numbers. Then, for ℜ(y)>ℜ(x)>0, one has
1. 2G1(w,x,y;z)=Γ(y)Γ(y−x)Γ(x)∫10sx−1(1−s)y−x−1(1−sz)−wds;
2. 2G1(w,x,y;z)= 2G1(x,w,y;z);
3. 2G1(w,x,y;z)=(1−z)−w2G1(w,y−x,y;zz−1).
Firstly, we derive the following result.
Theorem 1. Let 0≤α<1 and k≥11−α. If f∈k-Q(α), then f∈B(δ), where
δ=(2μ−λ)+√(2μ−λ)2+8λ4(λ=1+αkk(1−α);μ=k−αk−1k(1−α)). | (3.1) |
Proof. Let f′=ℏ, where ℏ is analytic in Δ with ℏ(0)=1. From inequality (1.5) which takes the form
ℜ(1+zℏ′(z)ℏ(z))>k|(1−α)ℏ(z)+α(1+zℏ′(z)ℏ(z))−1|=k|1−α−ℏ(z)+αℏ(z)−αzℏ′(z)ℏ(z)|, |
we find that
ℜ(ℏ(z)+1+αkk(1−α)zℏ(z)ℏ(z))>k−αk−1k(1−α), |
which can be rewritten as
ℜ(ℏ(z)+λzℏ(z)ℏ(z))>μ(λ=1+αkk(1−α);μ=k−αk−1k(1−α)). |
The above relationship can be written as the following Briot-Bouquet differential subordination
ℏ(z)+λzℏ′(z)ℏ(z)≺1+(1−2μ)z1−z. |
Thus, by Lemma 1, we obtain
ℏ≺1+(1−2δ)z1−z, | (3.2) |
where δ is given by (3.1). The relationship (3.2) implies that f∈B(δ). We thus complete the proof of Theorem 3.1.
Theorem 2. Let 0<α≤1, 0<β<1, c>0, k≥1, n≥m+1(m∈ N ), |ℓ|≥n2(c+1c) and
|αβℓ±(1−α)cβsinβπ2|≥1. | (3.3) |
If
f(z)=z+∞∑n=m+1anzn(am+1≠0) |
and f∈k-Q(α), then f∈B(β0), where
β0=min |
such that (3.3) holds.
Proof. By the assumption, we have
\begin{equation} f'(z) = \hslash(z) = 1+\mathop {\mathop \sum \limits^\infty }\limits_{n = m} c_{n}z^{n}\quad (c_{m}\neq0). \end{equation} | (3.4) |
In view of (1.5) and (3.4), we get
{\Re}\left(1+\frac{z\hslash'(z) }{\hslash(z)}\right) \gt k\left\vert \left(1-\alpha\right) \hslash(z)+\alpha\left(1+\frac{z\hslash'(z)}{\hslash(z)}\right) -1\right\vert . |
If there exists a point z_{0}\in\Delta such that
\left\vert \arg\hslash\left( z\right) \right\vert \lt \frac{\beta\pi} {2}\quad(\left\vert z\right\vert \lt \left\vert z_{0}\right\vert;\, 0 \lt \beta \lt 1) |
and
\left\vert \arg\hslash\left(z_{0}\right)\right\vert = \frac{\beta\pi}{2}\quad(0 \lt \beta \lt 1), |
then from Lemma 2, we know that
\frac{z_{0} \hslash'\left(z_{0}\right)}{\hslash\left(z_{0}\right) } = i\ell\beta, |
where
\left(\hslash\left(z_{0}\right)\right) ^{1/\beta} = \pm ic\quad\left(c \gt 0\right) |
and
\ell:\left\{ \begin{array} [c]{c} \ell\geq\frac{n}{2}\left(c+\frac{1}{c}\right)\quad (\arg\hslash\left(z_{0}\right) = \frac{\beta\pi}{2}), \\ \\ \ell\leq-\frac{n}{2}\left(c+\frac{1}{c}\right)\quad(\arg \hslash\left(z_{0}\right) = -\frac{\beta\pi}{2}). \end{array} \right. |
For the case
\arg\hslash\left(z_{0}\right) = \frac{\beta\pi}{2}, |
we get
\begin{equation} {\Re}\left(1+\frac{z_0\hslash'(z_0) }{\hslash(z_0)}\right) = {\Re}\left(1+i\ell \beta\right) = 1. \end{equation} | (3.5) |
Moreover, we find from (3.3) that
\begin{align} \begin{split} & k\left\vert\left(1-\alpha\right)\hslash(z_0) +\alpha\left(1+\frac{z_0\hslash'(z_0)}{\hslash(z_0)}\right) -1\right\vert \\ = &k\left\vert\left(1-\alpha\right)\left(\hslash(z_0) -1\right)+\alpha\frac{z_0\hslash'(z_0)}{\hslash(z_0)}\right\vert \\ = &k\left\vert\left(1-\alpha\right)\left[\left(\pm ic\right)^{\beta }-1\right]+i\alpha\beta\ell\right\vert \\ = &k\sqrt{\left(1-\alpha\right)^2\left(c^{\beta}\cos\frac{\beta\pi} {2}-1\right)^{2}+\left[\alpha\beta\ell\pm\left(1-\alpha\right)c^{\beta}\sin \frac{\beta\pi}{2}\right]^{2}}\\ \geq&1. \end{split} \end{align} | (3.6) |
By virtue of (3.5) and (3.6), we have
{\Re}\left(1+\frac{z\hslash'(z_0) }{\hslash(z_0)}\right)\leq k\left\vert \left(1-\alpha\right) \hslash(z_0)+\alpha\left(1+\frac{z_0\hslash(z_0)}{\hslash(z_0)}\right)-1\right\vert, |
which is a contradiction to the definition of k - \mathcal{Q}(\alpha) . Since \beta_{0} = {\min}\{\beta: \beta\in(0, 1)\} such that (3.3) holds, we can deduce that f\in\mathcal{B}(\beta_0) .
By using the similar method as given above, we can prove the case
\arg\hslash(z_{0}) = -\frac{\beta\pi}{2} |
is true. The proof of Theorem 2 is thus completed.
Theorem 3. If 0 < \beta < 1 and 0\leq\nu < 1 . If f\in k - \mathcal{Q}(\alpha) , then
{\Re}(f') \gt \left[ _{2}G_{1}\left(\frac{2}{\beta}\left( 1-\nu\right), 1;\frac{1}{\beta}+1;\frac{1}{2}\right)\right]^{-1}, |
or equivalently, k - \mathcal{Q}\left(\alpha\right)\subset{\mathcal{B}}\left(\nu_{0}\right) , where
\nu_{0} = \left[ _{2}G_{1}\left(\frac{2}{\beta}\left(1-\mu\right) , 1;\frac{1}{\beta}+1;\frac{1}{2}\right)\right]^{-1}. |
Proof. For
w = \frac{2}{\beta}(1-\nu), \ x = \frac {1}{\beta}, \ y = \frac{1}{\beta}+1, |
we define
\begin{align} \text{$\digamma$}(z) = \left(1+Dz\right)^{w}\int_0^1t^{x-1}\left(1+Dtz\right)^{-w}dt = \frac{\Gamma\left(x\right)}{\Gamma\left(y\right)}\ _{2} G_{1}\left(1, w, y;\frac{z}{z-1}\right). \end{align} | (3.7) |
To prove k - \mathcal{Q}(\alpha)\subset\mathcal{B}\left(\nu _{0}\right) , it suffices to prove that
\underset{\left\vert z\right\vert \lt 1}{\inf}\left\{{\Re}(q\left(z\right))\right\} = q\left(-1\right), |
which need to show that
{\Re}\left(1/\text{$\digamma$}(z)\right) \geq1/\text{$\digamma$}(-1). |
By Lemma 3 and (3.7), it follows that
\text{$\digamma$}(z) = \int_0^1\text{$\digamma$}\left(z, t\right)d\varepsilon(t), |
where
\begin{array}{l} \text{$\digamma$}(z, t) = \frac{1-z}{1-\left(1-t\right) z}\quad \left(0\leq t\leq1\right), \end{array} |
and
d\varepsilon(t) = \frac{\Gamma(x) } {\Gamma(w) \Gamma\left(y-w\right)}t^{w-1}\left(1-t\right) ^{y-w-1}dt, |
which is a positive measure on \left[0, 1\right] .
It is clear that {\Re}(\digamma(z, t)) > 0 and \digamma(-r, t) is real for \left\vert z\right\vert \leq r < 1 and t\in\left[0, 1\right] . Also
{\Re}\left(\frac{1}{\text{$\digamma$}(z, t) }\right) = {\Re}\left(\frac{1-\left(1-t\right)z} {1-z}\right)\geq\frac{1+\left(1-t\right)r}{1+r} = \frac{1} {\text{$\digamma$}(-r, t)} |
for \left\vert z\right\vert \leq r < 1 . Therefore, by Lemma 3, we get
{\Re}(1/\text{$\digamma$}(z)) \geq1/\text{$\digamma$}(-r). |
If we let r\rightarrow1^{-} , it follows that
{\Re}\left(1/\text{$\digamma$}(z)\right) \geq1/\text{$\digamma$}(-1). |
Thus, we deduce that k - \mathcal{Q}\left(\alpha\right)\subset\mathcal{B}(\nu_{0}) .
Theorem 4. Let 0\leq\alpha < 1 and k\geq\frac{1}{1-\alpha} . If f\in k - \mathcal{Q}\left(\alpha\right) , then
f'(z)\prec s(z) = \frac{1}{g(z)}, |
where
g(z) = {_{2}G_{1}\left(\frac{2}{\lambda}, 1, \frac{1}{\lambda}+1; \frac{z}{z-1}\right)}\quad\left(\lambda = \frac{1+\alpha k}{k(1-\alpha)}\right). |
Proof. Suppose that f' = \hslash . From the proof of Theorem 1, we see that
\hslash(z)+\frac{z\hslash'(z)} {\frac{1}{\lambda}\hslash(z)}\prec\frac{1+\left(1-2\mu \right)z}{1-z}\prec\frac{1+z}{1-z}\quad\left(\lambda = \frac{1+\alpha k}{k\left(1-\alpha\right)};\, \mu = \frac{k-\alpha k-1}{k(1-\alpha)}\right). |
If we set \lambda_1 = \frac{1}{\lambda} , \gamma_2 = 0, C = 1 and D = -1 in Lemma 4, then
\hslash(z)\prec s(z) = \frac{1}{g(z) } = \frac{z^{\frac{1}{\lambda}}\left(1-z\right)^{-\frac{2}{\lambda}}} {1/\lambda\int_0^z t^{(1/\lambda)-1}\left(1-t\right)^{-2/\lambda}dt}. |
By putting t = uz , and using Lemma 5, we obtain
\hslash(z)\prec s(z) = \frac{1}{g(z) } = \frac{1}{\frac{1}{\lambda}\left(1-z\right)^{\frac {2}{\lambda}}\int_0^1u^{(1/\lambda)-1}\left(1-uz\right)^{-2/\lambda}du} = \left[_{2}G_{1}\left(\frac{2}{\lambda}, 1, \frac {1}{\lambda}+1;\frac{z}{z-1}\right)\right]^{-1}, |
which is the desired result of Theorem 4.
The present investigation was supported by the Key Project of Education Department of Hunan Province under Grant no. 19A097 of the P. R. China. The authors would like to thank the referees for their valuable comments and suggestions, which was essential to improve the quality of this paper.
The authors declare no conflict of interest.
[1] |
Y. Bai, L. Gasiński, P. Winkert, S. D. Zeng, W1, p versus C1: The nonsmooth case involving critical growth, Bull. Math. Sci., 10 (2020), 2050009. https://doi.org/10.1142/S1664360720500095 doi: 10.1142/S1664360720500095
![]() |
[2] |
Y. Bai, S. Migórski, S. D. Zeng, A class of generalized mixed variational-hemivariational inequalities Ⅰ: Existence and uniqueness result, Comput. Math. Appl., 79 (2020), 2897–2911. https://doi.org/10.1016/j.camwa.2019.12.025 doi: 10.1016/j.camwa.2019.12.025
![]() |
[3] |
H. J. Brascamp, E. H. Lieb, J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal., 17 (1974), 227–237. https://doi.org/10.1016/0022-1236(74)90013-5 doi: 10.1016/0022-1236(74)90013-5
![]() |
[4] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006. |
[5] | Z. Lin, Z. Bai, Probability inequalities of random variables, Probability Inequalities, Springer, Berlin, Heidelberg, 2010, 37–50. https://doi.org/10.1007/978-3-642-05261-3_5 |
[6] | T. H. Zhao, O. Castillo, H. Jahanshahi, A. Yusuf, M. O. Alassafi, F. E. Alsaadi, et al., A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak, Appl. Comput. Math., 20 (2021), 160–176. |
[7] |
T. H. Zhao, M. K. Wang, Y. M. Chu, On the bounds of the perimeter of an ellipse, Acta Math. Sci., 42B (2022), 491–501. https://doi.org/10.1007/s10473-022-0204-y doi: 10.1007/s10473-022-0204-y
![]() |
[8] |
T. H. Zhao, M. K. Wang, G. J. Hai, Y. M. Chu, Landen inequalities for Gaussian hypergeometric function, RACSAM Rev. R. Acad. A, 116 (2022), 1–23. https://doi.org/10.1007/s13398-021-01197-y doi: 10.1007/s13398-021-01197-y
![]() |
[9] |
M. K. Wang, M. Y. Hong, Y. F. Xu, Z. H. Shen, Y. M. Chu, Inequalities for generalized trigonometric and hyperbolic functions with one parameter, J. Math. Inequal., 14 (2020), 1–21. https://doi.org/10.7153/jmi-2020-14-01 doi: 10.7153/jmi-2020-14-01
![]() |
[10] |
T. H. Zhao, W. M. Qian, Y. M. Chu, Sharp power mean bounds for the tangent and hyperbolic sine means, J. Math. Inequal., 15 (2021), 1459–1472. https://doi.org/10.7153/jmi-2021-15-100 doi: 10.7153/jmi-2021-15-100
![]() |
[11] |
M. U. Awan, N. Akhtar, S. Iftikhar, M. A. Noor, Y. M. Chu, New Hermite-Hadamard type inequalities for n-polynomial harmonically convex functions, J. Inequal. Appl., 2020 (2020). https://doi.org/10.1186/s13660-020-02393-x doi: 10.1186/s13660-020-02393-x
![]() |
[12] |
M. A. Latif, S. Rashi, S. S. Dragomir, Y. M. Chu, Hermite-Hadamard type inequalities for co-ordinated convex and quasi-convex functions and their applications, J. Inequal. Appl., 2019 (2019). https://doi.org/10.1186/s13660-019-2272-7 doi: 10.1186/s13660-019-2272-7
![]() |
[13] |
Y. M. Chu, G. D. Wang, X. H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetric function, Math. Nachr., 284 (2011), 53–663. https://doi.org/10.1002/mana.200810197 doi: 10.1002/mana.200810197
![]() |
[14] |
Y. M. Chu, W. F. Xia, X. H. Zhang, The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications, J. Multivariate Anal., 105 (2012), 412–42. https://doi.org/10.1016/j.jmva.2011.08.004 doi: 10.1016/j.jmva.2011.08.004
![]() |
[15] |
S. Z. Ullah, M. A. Khan, Z. A. Khan, Y. M. Chu, Integral majorization type inequalities for the functions in the sense of strong convexity, J. Funct. Space., 2019 (2019). https://doi.org/10.1155/2019/9487823 doi: 10.1155/2019/9487823
![]() |
[16] |
S. Z. Ullah, M. A. Khan, Y. M. Chu, Majorization theorems for strongly convex functions, J. Inequal. Appl., 2019 (2019). https://doi.org/10.1186/s13660-019-2007-9 doi: 10.1186/s13660-019-2007-9
![]() |
[17] | K. S. Zhang, J. P. Wan, p-convex functions and their properties, Pure Appl. Math., 23 (2007), 130–133. |
[18] |
S. Z. Ullah, M. A. Khan, Y. M. Chu, A note on generalized convex functions, J. Inequal. Appl., 2019 (2019). https://doi.org/10.1186/s13660-019-2242-0 doi: 10.1186/s13660-019-2242-0
![]() |
[19] |
S. N. Hajiseyedazizi, M. E. Samei, J. Alzabut, Y. M. Chu, On multi-step methods for singular fractional q-integro-differential equations, Open Math., 19 (2021), 1378–1405. https://doi.org/10.1515/math-2021-0093 doi: 10.1515/math-2021-0093
![]() |
[20] |
F. Jin, Z. S. Qian, Y. M. Chu, M. Rahman, On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative, J. Appl. Anal. Comput., 12 (2022), 790–806. https://doi.org/10.11948/20210357 doi: 10.11948/20210357
![]() |
[21] |
F. Z. Wang, M. N. Khan, I. Ahmad, H. Ahmad, H. Abu-Zinadah, Y. M. Chu, Numerical solution of traveling waves in chemical kinetics: Time-fractional fisher's equations, Fractals, 30 (2022), 2240051. https://doi.org/10.1142/S0218348X22400515 doi: 10.1142/S0218348X22400515
![]() |
[22] |
T. H. Zhao, B. A. Bhayo, Y. M. Chu, Inequalities for generalized Grötzsch ring function, Comput. Meth. Funct. Th., 22 (2022), 559–574. https://doi.org/10.1007/s40315-021-00415-3 doi: 10.1007/s40315-021-00415-3
![]() |
[23] |
M. B. Khan, M. A. Noor, M. M. Al-Shomrani, L. Abdullah, Some novel inequalities for LR-h-convex interval-valued functions by means of pseudo order relation, Math. Meth. Appl. Sci., 45 (2022), 1310–1340. https://doi.org/10.1002/mma.7855 doi: 10.1002/mma.7855
![]() |
[24] |
M. B. Khan, J. E. Macías-Díaz, S. Treanta, M. S. Soliman, H. G. Zaini, Hermite-Hadamard inequalities in fractional calculus for left and right harmonically convex functions via interval-valued settings, Fractal Fract., 6 (2022), 178. https://doi.org/10.3390/fractalfract6040178 doi: 10.3390/fractalfract6040178
![]() |
[25] |
M. B. Khan, M. A. Noor, J. E. Macías-Díaz, M. S. Soliman, H. G. Zaini, Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation, Demonstr. Math., 55 (2022), 387–403. https://doi.org/10.1515/dema-2022-0023 doi: 10.1515/dema-2022-0023
![]() |
[26] |
T. H. Zhao, Z. Y. He, Y. M. Chu, Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals, Comput. Meth. Funct. Th., 21 (2021), 413–426. https://doi.org/10.1007/s40315-020-00352-7 doi: 10.1007/s40315-020-00352-7
![]() |
[27] |
T. H. Zhao, M. K. Wang, Y. M. Chu, Concavity and bounds involving generalized elliptic integral of the first kind, J. Math. Inequal., 15 (2021), 701–724. https://doi.org/10.7153/jmi-2021-15-50 doi: 10.7153/jmi-2021-15-50
![]() |
[28] |
T. H. Zhao, M. K. Wang, Y. M. Chu, Monotonicity and convexity involving generalized elliptic integral of the first kind, RACSAM Rev. R. Acad. A, 115 (2021), 1–13. https://doi.org/10.1007/s13398-020-00992-3 doi: 10.1007/s13398-020-00992-3
![]() |
[29] |
H. H. Chu, T. H. Zhao, Y. M. Chu, Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contra harmonic means, Math. Slovaca, 70 (2020), 1097–1112. https://doi.org/10.1515/ms-2017-0417 doi: 10.1515/ms-2017-0417
![]() |
[30] |
T. H. Zhao, Z. Y. He, Y. M. Chu, On some refinements for inequalities involving zero-balanced hyper geometric function, AIMS Math., 5 (2020), 6479–6495. https://doi.org/10.3934/math.2020418 doi: 10.3934/math.2020418
![]() |
[31] |
T. H. Zhao, M. K. Wang, Y. M. Chu, A sharp double inequality involving generalized complete elliptic integral of the first kind, AIMS Math., 5 (2020), 4512–4528. https://doi.org/10.3934/math.2020290 doi: 10.3934/math.2020290
![]() |
[32] |
T. H. Zhao, L. Shi, Y. M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, RACSAM Rev. R. Acad. A, 114 (2020), 1–14. https://doi.org/10.1007/s13398-020-00825-3 doi: 10.1007/s13398-020-00825-3
![]() |
[33] |
T. H. Zhao, B. C. Zhou, M. K. Wang, Y. M. Chu, On approximating the quasi-arithmetic mean, J. Inequal. Appl., 2019 (2019), 42. https://doi.org/10.1186/s13660-019-1991-0 doi: 10.1186/s13660-019-1991-0
![]() |
[34] |
T. H. Zhao, M. K. Wang, W. Zhang, Y. M. Chu, Quadratic transformation inequalities for Gaussian hyper geometric function, J. Inequal. Appl., 2018 (2018), 251. https://doi.org/10.1186/s13660-018-1848-y doi: 10.1186/s13660-018-1848-y
![]() |
[35] |
M. A. Khan, J. Pečarić, Y. M. Chu, Refinements of Jensen's and McShane's inequalities with applications, AIMS Math., 5 (2020), 4931–4945. https://doi.org/10.3934/math.2020315 doi: 10.3934/math.2020315
![]() |
[36] |
S. Khan, M. A. Khan, Y. M. Chu, Converses of the Jensen inequality derived from the Green functions with applications in information theory, Math. Meth. Appl. Sci., 43 (2020), 2577–2587. https://doi.org/10.1002/mma.6066 doi: 10.1002/mma.6066
![]() |
[37] |
Y. Sawano, H. Wadade, On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Orrey space, J. Fourier Anal. Appl., 19 (2013), 20–47. https://doi.org/10.1007/s00041-012-9223-8 doi: 10.1007/s00041-012-9223-8
![]() |
[38] |
P. Ciatti, M. G. Cowling, F. Ricci, Hardy and uncertainty inequalities on stratified Lie groups, Adv. Math., 277 (2015), 365–387. https://doi.org/10.1016/j.aim.2014.12.040 doi: 10.1016/j.aim.2014.12.040
![]() |
[39] | B. Gavrea, I. Gavrea, On some Ostrowski type inequalities, Gen. Math., 18 (2010), 33–44. |
[40] |
H. Gunawan, Fractional integrals and generalized Olsen inequalities, Kyungpook Math. J., 49 (2009), 31–39. https://doi.org/10.5666/KMJ.2009.49.1.031 doi: 10.5666/KMJ.2009.49.1.031
![]() |
[41] | J. Hadamard, Étude sur les propriétés des fonctions entières en particulier d'une fonction considérée par Riemann, J. Math. Pure Appl., 58 (1893), 171–215. |
[42] | L. Fejér, Uberdie Fourierreihen Ⅱ, Math. Naturwise. Anz Ungar. Akad. Wiss., 24 (1906), 369–390. |
[43] | R. E. Moore, Interval analysis, Prentice Hall, Englewood Cliffs, 1966. |
[44] | U. Kulish, W. Miranker, Computer arithmetic in theory and practice, Academic Press, New York, 2014. |
[45] |
D. Zhao, T. An, G. Ye, W. Liu, New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), 1–14. https://doi.org/10.1186/s13660-017-1594-6 doi: 10.1186/s13660-017-1594-6
![]() |
[46] | B. Bede, Studies in fuzziness and soft computing, Math of fuzzy sets fuzzy logic, Springer, Berlin/Heidelberg, 295 (2013). https://doi.org/10.1007/978-3-642-35221-8 |
[47] | W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Räumen, Pupl. Inst. Math., 23 (1978), 13–20. |
[48] |
Y. Chalco-Cano, A. Flores-Franulič, H. Román-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457–472. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608617 doi: 10.1109/IFSA-NAFIPS.2013.6608617
![]() |
[49] |
Y. Chalco-Cano, W. A. Lodwick, W. Condori-Equice, Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput., 19 (2015), 3293–3300. https://doi.org/10.1007/s00500-014-1483-6 doi: 10.1007/s00500-014-1483-6
![]() |
[50] |
T. M. Costa, H. Román-Flores, Y. Chalco-Cano, Opial-type inequalities for interval-valued functions, Fuzzy Set. Syst., 358 (2019), 48–63. https://doi.org/10.1016/j.fss.2018.04.012 doi: 10.1016/j.fss.2018.04.012
![]() |
[51] | S. S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335–341. |
[52] |
Z. B. Fang, R. J. Shi, On the (p, h)-convex function and some integral inequalities, J. Inequal. Appl., 2014 (2014). https://doi.org/10.1186/1029-242X-2014-13 doi: 10.1186/1029-242X-2014-13
![]() |
[53] |
M. Kunt, İ. İşcan, Hermite-Hadamard-Fejer type inequalities for p-convex functions, Arab J. Math. Sci., 23 (2017), 215–230. https://doi.org/10.1016/j.ajmsc.2016.11.001 doi: 10.1016/j.ajmsc.2016.11.001
![]() |
[54] |
H. Román-Flores, Y. Chalco-Cano, W. A. Lodwick, Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37 (2018), 1306–1318. https://doi.org/10.1007/s40314-016-0396-7 doi: 10.1007/s40314-016-0396-7
![]() |
[55] |
S. Varošanec, On h-convexity, J. Math. Anal. Appl., 326 (2007), 303–311. https://doi.org/10.1016/j.jmaa.2006.02.086 doi: 10.1016/j.jmaa.2006.02.086
![]() |
[56] |
Y. M. Chu, T. H. Zhao, Concavity of the error function with respect to Hölder means, Math. Inequal. Appl., 19 (2016), 589–595. https://doi.org/10.7153/mia-19-43 doi: 10.7153/mia-19-43
![]() |
[57] |
W. M. Qian, H. H. Chu, M. K. Wang, Y. M. Chu, Sharp inequalities for the Toader mean of order –1 in terms of other bivariate means, J. Math. Inequal., 16 (2022), 127–141. https://doi.org/10.7153/jmi-2022-16-10 doi: 10.7153/jmi-2022-16-10
![]() |
[58] |
T. H. Zhao, H. H. Chu, Y. M. Chu, Optimal Lehmer mean bounds for the nth power-type Toader mean of n = −1, 1, 3, J. Math. Inequal., 16 (2022), 157–168. https://doi.org/10.7153/jmi-2022-16-12 doi: 10.7153/jmi-2022-16-12
![]() |
[59] |
T. H. Zhao, M. K. Wang, Y. Q. Dai, Y. M. Chu, On the generalized power-type Toader mean, J. Math. Inequal., 16 (2022), 247–264. https://doi.org/10.7153/jmi-2022-16-18 doi: 10.7153/jmi-2022-16-18
![]() |
[60] |
M. B. Khan, T. Savin, H. Alrweili, T. Saeed, M. S. Soliman, Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings, AIMS Math., 7 (2022), 15659–15679. https://doi.org/10.3934/math.2022857 doi: 10.3934/math.2022857
![]() |
[61] |
M. B. Khan, O. M. Alsalami, S. Treanțǎ, T. Saeed, K. Nonlaopon, New class of convex interval-valued functions and Riemann Liouville fractional integral inequalities, AIMS Math., 7 (2022), 15497–15519. https://doi.org/10.3934/math.2022849 doi: 10.3934/math.2022849
![]() |
[62] |
T. Saeed, M. B. Khan, S. Treanțǎ, H. H. Alsulami, M. S. Alhodaly, Interval Fejér-type inequalities for left and right-λ-preinvex functions in interval-valued settings, Axioms, 11 (2022), 368. https://doi.org/10.3390/axioms11080368 doi: 10.3390/axioms11080368
![]() |
[63] |
M. B. Khan, A. Cătaş, O. M. Alsalami, Some new estimates on coordinates of generalized convex interval-valued functions, Fractal Fract., 6 (2022), 415. https://doi.org/10.3390/fractalfract6080415 doi: 10.3390/fractalfract6080415
![]() |
[64] |
L. A. Zadeh, Fuzzy sets, Inform. Cont., 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
![]() |
[65] |
S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets Syst., 48 (1992), 129–132. https://doi.org/10.1016/0165-0114(92)90256-4 doi: 10.1016/0165-0114(92)90256-4
![]() |
[66] |
S. S. Chang, Y. G. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Set. Syst., 32 (1989), 359–367. https://doi.org/10.1016/0165-0114(89)90268-6 doi: 10.1016/0165-0114(89)90268-6
![]() |
[67] |
M. A. Noor, Fuzzy preinvex functions, Fuzzy Set. Syst., 64 (1994), 95–104. https://doi.org/10.1016/0165-0114(94)90011-6 doi: 10.1016/0165-0114(94)90011-6
![]() |
[68] |
B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Set. Syst., 151 (2005), 581–599. https://doi.org/10.1016/j.fss.2004.08.001 doi: 10.1016/j.fss.2004.08.001
![]() |
[69] |
A. Ben-Isreal, B. Mond, What is invexity? Anziam J., 1986, 1–9. https://doi.org/10.1017/S0334270000005142 doi: 10.1017/S0334270000005142
![]() |
[70] |
Y. Chalco-Cano, M. A. Rojas-Medar, H. Román-Flores, M-convex fuzzy mappings and fuzzy integral mean, Comput. Math. Appl., 40 (2000), 1117–1126. https://doi.org/10.1016/S0898-1221(00)00226-1 doi: 10.1016/S0898-1221(00)00226-1
![]() |
[71] | P. Diamond, P. E. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientific, 1994. https://doi.org/10.1142/2326 |
[72] |
J. R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Set. Syst., 18 (1986), 31–43. https://doi.org/10.1016/0165-0114(86)90026-6 doi: 10.1016/0165-0114(86)90026-6
![]() |
[73] |
O. Kaleva, Fuzzy differential equations, Fuzzy Set. Syst., 24 (1987), 301–317. https://doi.org/10.1016/0165-0114(87)90029-7 doi: 10.1016/0165-0114(87)90029-7
![]() |
[74] |
M. L. Puri, D. A. Ralescu, Fuzzy random variables, Read. Fuzzy Set. Intell. Syst., 114 (1986), 409–422. https://doi.org/10.1016/0022-247X(86)90093-4 doi: 10.1016/0022-247X(86)90093-4
![]() |
[75] |
S. A. Iqbal, M. G. Hafez, Y. M. Chu, C. Park, Dynamical analysis of nonautonomous RLC circuit with the absence and presence of Atangana-Baleanu fractional derivative, J. Appl. Anal. Comput., 12 (2022), 770–789. https://doi.org/10.11948/20210324 doi: 10.11948/20210324
![]() |
[76] |
T. R. Huang, L. Chen, Y. M. Chu, Asymptotically sharp bounds for the complete p-elliptic integral of the first kind, Hokkaido Math. J., 51 (2022), 189–210. https://doi.org/10.14492/hokmj/2019-212 doi: 10.14492/hokmj/2019-212
![]() |
[77] |
T. H. Zhao, W. M. Qian, Y. M. Chu, On approximating the arc lemniscate functions, Indian J. Pure Appl. Math., 53 (2022), 316–329. https://doi.org/10.1007/s13226-021-00016-9 doi: 10.1007/s13226-021-00016-9
![]() |
[78] |
G. Santos-García, M. B. Khan, H. Alrweili, A. A. Alahmadi, S. S. Ghoneim, Hermite-Hadamard and Pachpatte type inequalities for coordinated preinvex fuzzy-interval-valued functions pertaining to a fuzzy-interval double integral operator, Mathematics, 10 (2022), 2756. https://doi.org/10.3390/math10152756 doi: 10.3390/math10152756
![]() |
[79] |
J. E. Macías-Díaz, M. B. Khan, H. Alrweili, M. S. Soliman, Some fuzzy inequalities for harmonically s-convex fuzzy number valued functions in the second sense integral, Symmetry, 14 (2022), 1639. https://doi.org/10.3390/sym14081639 doi: 10.3390/sym14081639
![]() |
[80] |
M. B. Khan, M. A. Noor, H. G. Zaini, G. Santos-García, M. S. Soliman, The new versions of Hermite-Hadamard inequalities for pre-invex fuzzy-interval-valued mappings via fuzzy Riemann integrals, Int. J. Comput. Intell. Syst., 15 (2022), 66. https://doi.org/10.1007/s44196-022-00127-z doi: 10.1007/s44196-022-00127-z
![]() |
[81] |
M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, New Hermite-Hadamard type inequalities for (h1, h2)-convex fuzzy-interval-valued functions, Adv. Differ. Equ., 2021 (2021), 6–20. https://doi.org/10.1186/s13662-020-03166-y doi: 10.1186/s13662-020-03166-y
![]() |
[82] |
M. B. Khan, M. A. Noor, L. Abdullah, Y. M. Chu, Some new classes of preinvex fuzzy-interval-valued functions and inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1403–1418. https://doi.org/10.2991/ijcis.d.210409.001 doi: 10.2991/ijcis.d.210409.001
![]() |
[83] |
P. Liu, M. B. Khan, M. A. Noor, K. I. Noor, New Hermite-Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense, Complex Intell. Syst., 8 (2022), 413–427. https://doi.org/10.1007/s40747-021-00379-w doi: 10.1007/s40747-021-00379-w
![]() |
[84] |
G. Sana, M. B. Khan, M. A. Noor, P. O. Mohammed, Y. M. Chu, Harmonically convex fuzzy-interval-valued functions and fuzzy-interval Riemann-Liouville fractional integral inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1809–1822. https://doi.org/10.2991/ijcis.d.210620.001 doi: 10.2991/ijcis.d.210620.001
![]() |
[85] |
M. B. Khan, S. Treanțǎ, H. Budak, Generalized p-convex fuzzy-interval-valued functions and inequalities based upon the fuzzy-order relation, Fractal Fract., 6 (2022), 63. https://doi.org/10.3390/fractalfract6020063 doi: 10.3390/fractalfract6020063
![]() |
[86] | R. Osuna-Gómez, M. D. Jiménez-Gamero, Y. Chalco-Cano, M. A. Rojas-Medar, Hadamard and Jensen inequalities for s-convex fuzzy processes, Soft Methodology and Random Information Systems, Advances in Soft Computing, Springer, Berlin, Heidelberg, l26 (2004), 1–15. https://doi.org/10.1007/978-3-540-44465-7_80 |
[87] |
T. M. Costa, Jensen's inequality type integral for fuzzy-interval-valued functions, Fuzzy Set. Syst., 327 (2017), 31–47. https://doi.org/10.1016/j.fss.2017.02.001 doi: 10.1016/j.fss.2017.02.001
![]() |
[88] |
T. M. Costa, H. Roman-Flores, Some integral inequalities for fuzzy-interval-valued functions, Inform. Sci., 420 (2017), 110–125. https://doi.org/10.1016/j.ins.2017.08.055 doi: 10.1016/j.ins.2017.08.055
![]() |
[89] |
M. B. Khan, M. A. Noor, K. I. Noor, K. S. Nisar, K. A. Ismail, A. Elfasakhany, Some inequalities for LR-(h1, h2)-convex interval-valued functions by means of pseudo order relation, Int. J. Comput. Intell. Syst., 14 (2021), 1–15. https://doi.org/10.1007/s44196-021-00032-x doi: 10.1007/s44196-021-00032-x
![]() |
[90] |
M. B. Khan, M. A. Noor, H. M. Al-Bayatti, K. I. Noor, Some new inequalities for LR-log-h-convex interval-valued functions by means of pseudo order relation, Appl. Math., 15 (2021), 459–470. https://doi.org/10.18576/amis/150408 doi: 10.18576/amis/150408
![]() |
[91] |
M. B. Khan, M. A. Noor, T. Abdeljawad, A. A. A. Mousa, B. Abdalla, S. M. Alghamdi, LR-preinvex interval-valued functions and Riemann-Liouville fractional integral inequalities, Fractal Fract., 5 (2021), 243. https://doi.org/10.3390/fractalfract5040243 doi: 10.3390/fractalfract5040243
![]() |
[92] |
J. E. Macías-Díaz, M. B. Khan, M. A. Noor, A. M. A. Allah, S. M. Alghamdi, Hermite-Hadamard inequalities for generalized convex functions in interval-valued calculus, AIMS Math., 7 (2022), 4266–4292. https://doi.org/10.3934/math.2022236 doi: 10.3934/math.2022236
![]() |
[93] |
M. B. Khan, H. G. Zaini, S. Treanțǎ, M. S. Soliman, K. Nonlaopon, Riemann-Liouville fractional integral inequalities for generalized pre-invex functions of interval-valued settings based upon pseudo order relation, Mathematics, 10 (2022), 204. https://doi.org/10.3390/math10020204 doi: 10.3390/math10020204
![]() |
[94] |
M. B. Khan, S. Treanțǎ, M. S. Soliman, K. Nonlaopon, H. G. Zaini, Some Hadamard-Fejér type inequalities for LR-convex interval-valued functions, Fractal Fract., 6 (2022), 6. https://doi.org/10.3390/fractalfract6010006 doi: 10.3390/fractalfract6010006
![]() |
[95] |
M. B. Khan, G. Santos-García, M. A. Noor, M. S. Soliman, Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities, Chaos Soliton. Fract., 164 (2022), 112692. https://doi.org/10.1016/j.chaos.2022.112692 doi: 10.1016/j.chaos.2022.112692
![]() |
[96] |
Z. H. Liu, D. Motreanu, S. D. Zeng, Generalized penalty and regularization method for differential variational- hemivariational inequalities, SIAM J. Optim., 31 (2021), 1158–1183. https://doi.org/10.1137/20M1330221 doi: 10.1137/20M1330221
![]() |
[97] |
Y. J. Liu, Z. H. Liu, C. F. Wen, J. C. Yao, S. D. Zeng, Existence of solutions for a class of noncoercive variational-hemivariational inequalities arising in contact problems, Appl. Math. Optim., 84 (2021), 2037–2059. https://doi.org/10.1007/s00245-020-09703-1 doi: 10.1007/s00245-020-09703-1
![]() |
[98] |
S. D. Zeng, S. Migorski, Z. H. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, SIAM J. Optim., 31 (2021), 2829–2862. https://doi.org/10.1137/20M1351436 doi: 10.1137/20M1351436
![]() |
[99] |
Y. J. Liu, Z. H. Liu, D. Motreanu, Existence and approximated results of solutions for a class of nonlocal elliptic variational-hemivariational inequalities, Math. Method. Appl. Sci., 43 (2020), 9543–9556. https://doi.org/10.1002/mma.6622 doi: 10.1002/mma.6622
![]() |
[100] |
Y. J. Liu, Z. H. Liu, C. F. Wen, Existence of solutions for space-fractional parabolic hemivariational inequalities, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1297–1307. https://doi.org/10.3934/dcdsb.2019017 doi: 10.3934/dcdsb.2019017
![]() |
[101] |
Z. H. Liu, N.V. Loi, V. Obukhovskii, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Int. J. Bifurcat. Chaos Appl. Sci. Eng., 23 (2013), 1350125. https://doi.org/10.1142/S0218127413501253 doi: 10.1142/S0218127413501253
![]() |
1. | Syed Ghoos Ali Shah, Saima Noor, Saqib Hussain, Asifa Tasleem, Akhter Rasheed, Maslina Darus, Rashad Asharabi, Analytic Functions Related with Starlikeness, 2021, 2021, 1563-5147, 1, 10.1155/2021/9924434 |