Hepatitis is an infectious disease typified by inflammation in internal organ tissues, and it is caused by infection or inflammation of the liver. Hepatitis is often feared as a fatal illness, especially in developing countries, mostly due to contaminated water, poor sanitation, and risky blood transfusion practices. Although viruses are typically blamed, other potential causes of this kind of liver infection include autoimmune disorders, toxins, medicines, opioids, and alcohol. Viral hepatitis may be diagnosed using a variety of methods, including a physical exam, liver surgery (biopsy), imaging investigations like an ultrasound or CT scan, blood tests, a viral serology panel, a DNA test, and viral antibody testing. Our study proposes a new decision-support system for hepatitis diagnosis based on spherical q-linear Diophantine fuzzy sets (Sq-LDFS). Sq-LDFS form the generalized structure of all existing notions of fuzzy sets. Furthermore, a list of novel Einstein aggregation operators is developed under Sq-LDF information. Also, an improved VIKOR method is presented to address the uncertainty in analyzing the viral hepatitis categories demonstration. Interesting and useful properties of the proposed operators are given. The core of this research is the proposed algorithm based on the proposed Einstein aggregation operators and improved VIKOR approach to address uncertain information in decision support problems. Finally, a hepatitis diagnosis case study is examined to show how the suggested approach works in practice. Additionally, a comparison is provided to demonstrate the superiority and efficacy of the suggested decision technique.
Citation: Huzaira Razzaque, Shahzaib Ashraf, Wajdi Kallel, Muhammad Naeem, Muhammad Sohail. A strategy for hepatitis diagnosis by using spherical q-linear Diophantine fuzzy Dombi aggregation information and the VIKOR method[J]. AIMS Mathematics, 2023, 8(6): 14362-14398. doi: 10.3934/math.2023735
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Hepatitis is an infectious disease typified by inflammation in internal organ tissues, and it is caused by infection or inflammation of the liver. Hepatitis is often feared as a fatal illness, especially in developing countries, mostly due to contaminated water, poor sanitation, and risky blood transfusion practices. Although viruses are typically blamed, other potential causes of this kind of liver infection include autoimmune disorders, toxins, medicines, opioids, and alcohol. Viral hepatitis may be diagnosed using a variety of methods, including a physical exam, liver surgery (biopsy), imaging investigations like an ultrasound or CT scan, blood tests, a viral serology panel, a DNA test, and viral antibody testing. Our study proposes a new decision-support system for hepatitis diagnosis based on spherical q-linear Diophantine fuzzy sets (Sq-LDFS). Sq-LDFS form the generalized structure of all existing notions of fuzzy sets. Furthermore, a list of novel Einstein aggregation operators is developed under Sq-LDF information. Also, an improved VIKOR method is presented to address the uncertainty in analyzing the viral hepatitis categories demonstration. Interesting and useful properties of the proposed operators are given. The core of this research is the proposed algorithm based on the proposed Einstein aggregation operators and improved VIKOR approach to address uncertain information in decision support problems. Finally, a hepatitis diagnosis case study is examined to show how the suggested approach works in practice. Additionally, a comparison is provided to demonstrate the superiority and efficacy of the suggested decision technique.
In this paper, we primarily focus on investigating different properties of generalized (p,q)-elliptic integrals and the generalized (p,q)-Hersch-Pfluger distortion function. In recent years, mathematicians have made significant progress in studying inequalities and various properties related to complete elliptic integrals, especially Legendre elliptic integrals and generalized complete elliptic integrals of the first and second types [3,4,5,10,11,23,26].
We first introduce some necessary notation. For complex numbers a,b,c with c≠0,−1,−2,…, and x∈(−1,1), the Gaussian hypergeometric function [9] is defined as follows:
F(a,b;c;x)=2F1(a,b;c;x)=∞∑n=0(a,n)(b,n)(c,n)xnn!, | (1.1) |
where (a,n)≡a(a+1)⋯(a+n−1) is the shifted factorial function for n∈N+, and (a,0)=1 for a≠0. As we all know, 2F1(a,b;c;x) has many important applications in the theory of geometric functions and several other contexts [19]. Many special functions in mathematical physics are special or limit cases of this function [1]. Bhayo studied a new form of the generalized (p,q)-complete elliptic integrals as an application of generalized (p,q)-trigonometric functions [6]. In recent years, the generalization of classical trigonometric functions has attracted significant interest [8,20]. For this, we need the generalized arcsine function arcsinp,q(x) and the generalized πp,q. For p,q∈(1,∞), set
arcsinp,q(x)≡∫x0dt(1−tq)1/p,x∈[0,1], |
and the generalized πp,q is the number defined by
πp,q=2arcsinp,q(1)≡2∫10dt(1−tq)1/p=2qB(1−1p,1q), |
where B is the beta function. For Rex>0 and Rey>0, the classical gamma function Γ(x) and beta function B(x,y) are respectively defined as
Γ(x)=∫∞0e−ttx−1dt,B(x,y)=Γ(x)Γ(y)Γ(x+y). |
Clearly, arcsinp,q(x) is an increasing homeomorphism from [0,1] onto [0,πp,q/2], and its inverse function is the generalized (p,q)-sine function sinp,q is defined on the interval [0,πp,q/2]. Moreover, the function sinp,q can be extended to the interval [0,πp,q] by
sinp,q(x)=sinp,q(πp,q−x),x∈[πp,q/2,π]. |
sinp,q can be also extended to the whole R, and the generalized (p,q)-sine function reduces to the classical sine function for p=q=2.
Applying the definitions of sinp,q(x) and πp,q, we can define the generalized (p,q)-elliptic integrals of the first kind Kp,q and of the second kind Ep,q by
Kp,q(r)=∫πp,q/20dt(1−rqsinqp,q(t))1−1/p=∫10dt(1−tq)1/p(1−rqtq)1−1/p |
and
Ep,q(r)=∫πp,q/20(1−rqsinqp,q(t))1/pdt=∫10(1−rqtq1−tq)1/pdt |
respectively, for p,q∈(1,∞), r∈(0,1).
As a special case of the Gaussian hypergeometric function, these generalized (p,q)-elliptic integrals can be represented by Gaussian hypergeometric functions [15] as
{Kp,q=Kp,q(r)=πp,q22F1(1−1p,1q;1−1p+1q;rq)K′p,q=K′p,q(r)=Kp,q(r′)Kp,q(0)=πp,q2,Kp,q(1)=∞ | (1.2) |
and
{Ep,q=Ep,q(r)=πp,q22F1(−1p,1q;1−1p+1q;rq)E′p,q=E′p,q(r)=Ep,q(r′)Ep,q(0)=πp,q2,Ep,q(1)=1, | (1.3) |
where p,q∈(1,∞), r∈(0,1), r′=(1−rq)1/q. If p=q=2, we can derive the classical complete elliptic integrals K and E, which are well-known complete elliptic integrals of the first kind and second kind, respectively. These complete elliptic integrals play an important role in many branches of quasiconformal mapping, complex analysis, and physics.
In 2017, Yang et al. [24] showed that the ratio K(r)/ln(c/r′) is strictly concave if and only if c=e4/3 on (0,1), and K(r)/ln(1+4/r′) is strictly convex on (0,1).
In 2019, Wang et al. [22] presented the convexity of the function (E′a−r2K′a)/r′2 and some properties of the function αKa(rα) with respect to the parameter α.
In 2020, Huang et al. [13] established monotonicity properties for certain functions involving the complete p-elliptic integrals of the first and second kinds. They also presented the inequality π/2−log2+log(1+1/r′)+α(1−r′)<Kp(r)<π/2−log2+log(1+1/r′)+β(1−r′) which holds for all r∈(0,1) with the best possible constants α and β. Moreover, these generalized elliptic integrals have significant applications in the theory of geometric functions and in the theory of mean values. More properties and applications of these integrals are given in [7,13,17,21,22,24,25].
The generalized (p,q)-elliptic integrals of the first kind Kp,q and of the second kind Ep,q satisfy the following Legendre relation:
Kp,q(r)E′p,q(r)+K′p,q(r)Ep,q(r)−Kp,q(r)K′p,q(r)=πp,q2. | (1.4) |
This relation has important applications in many areas of mathematics and physics, including celestial mechanics, quantum mechanics, and statistical mechanics. when p=q=2, the equation reduces to the classical Legendre relation.
Inspired by the papers [22], [24] and [13], we can consider extending the results of K(r), Ka(r) and Kp(r) to the generalized (p,q)-elliptic integrals of the first kind Kp,q.
A generalized (p,q)-modular equation of degree h>0 is
2F1(a,b;c;1−sq)2F1(a,b;c;sq)=h2F1(a,b;c;1−rq)2F1(a,b;c;rq),r∈(0,1), | (1.5) |
which a,b,c>0 with a+b≥c. Using the decreasing homeomorphism μp,q:(0,1)→(0,∞) defined by
μp,q(r)=πp,q2K′p,q(r)Kp,q(r), |
for p,q∈(1,∞). The function μp,q is called the generalized (p,q)-Gr¨otzsch ring function, we can rewrite (1.5) as
μp,q(s)=hμp,q(r),r∈(0,1). | (1.6) |
The solution of (1.6) is given by
s=φp,qK(r)=μ−1p,q(μp,q(r)/K). | (1.7) |
For p,q∈(1,∞), r∈(0,1), K∈(0,∞), we have
φp,qK(r)q+φp,q1/K(r′)q=1. | (1.8) |
The function φp,qK(r) is referred to as the generalized (p,q)-Hersch-Pfluger distortion function with degree K=1/h. For p=q=2, the functions μp,q(r) and φp,qK(r) reduces to well-known special cases that Gr¨otzsch ring function μ(r) and Hersch-Pfluger distortion function φK(r), respectively, which play important role in the theory of plane quasiconformal mappings.
In 2015, Alzer et al. [2] studied the monotonicity, convexity, and concavity properties of the function μ(rα)/α, and established various Gr¨otzsch ring functional inequalities based on these properties.
In [3], the authors focused on studying the properties of the generalized Gr¨otzsch ring function μa(r) and the generalized Hersch-Pfluger distortion function φaK(r). They derived various inequalities involving μa(r) and r−1/KφaK(r) by utilizing the monotonicity, concavity, and convexity of the functions μa(r), r−1/KφaK(r), log(1/φaK(e−x)), and φaK(rx)/φaK(x).
In 2022, Lin et al. [16] explored the monotonicity and convexity properties of the function μp,q(r) and obtained sharp functional inequalities that sharpen and extend some existing results on the modulus of μ(r).
Inspired by the papers [3], [12], and [16], our motivation is to extend the existing results of the functions φK(r) and φaK(r) to the generalized (p,q)-Hersch-Pfluger distortion function. Our goal is to gain the properties of the generalized (p,q)-Hersch-Pfluger distortion function and derive sharp functional inequalities for this function.
Our main objective of this paper is to investigate various properties of the generalized (p,q)-elliptic integrals and the generalized (p,q)-Hersch-Pfluger distortion function. We specifically focus on establishing complete monotonicity, logarithmic, geometric concavity, and convexity properties of certain functions involving these generalized integrals and arcsine functions. Additionally, they derive several sharp functional inequalities for the generalized (p,q)-Hersch-Pfluger distortion function, which improve upon and generalize existing results. Apart from the introduction, this paper consists of three additional sections. Section 2 contains some preliminaries as well as several formulas and lemmas. In Section 3, we present some of the major results regarding the generalized (p, q)-elliptic integrals and provide their proof. In Section 4, we study the generalized (p, q)-Hersch-Pfluger distortion function, and present some of the major results and provide their proof.
In this section, we present several formulas and lemmas that have been extensively utilized in the paper. These formulas and lemmas play a crucial role in the analysis and proofs of the major results. Throughout this paper, we denote p,q∈(1,∞), r∈(0,1) and r′=(1−rq)1/q.
Lemma 2.1. [16] Derivative formulas:
(1)dKp,qdr=Ep,q−r′qKp,qrr′q,(2)dEp,qdr=q(Ep,q−Kp,q)pr,(3)dμp,q(r)dr=−π2p,q4rr′qK2p,q,(4)d(Ep,q−r′qKp,q)dr=(p−q)(Kp,q−Ep,q)+p(q−1)rqKp,qpr,(5)d(Kp,q−Ep,q)dr=(p−qr′q)Ep,q+(q−p)r′qKp,qprr′q. |
Based on the derivative formula Lemma 2.1, we derive the derivative formulas of the function φp,qK(r) in the following lemma.
Lemma 2.2. Let p,q∈(1,∞), then
∂φp,qK(r)∂r=ss′qKp,q(s)K′p,q(s)rr′qKp,q(r)K′p,q(r)=1Kss′qKp,q(s)2rr′qKp,q(r)2=Kss′qK′p,q(s)2rr′qK′p,q(r)2 |
for r∈(0,1).
Proof. By the definitions of s=φp,qK(r), μp,q(s)=μp,q(r)/K, and the derivative formulas of Kp,q(r), we can derive the following equation
−π2p,q4ss′qKp,q(s)2⋅∂s∂r=−1Kπ2p,q4rr′qKp,q(r)2, |
then the derivative formulas of φp,qK(r) is following.
Lemma 2.3. [4] For p∈[0,∞), let I=[0,p), and suppose that f,g:I→[0,∞) are functions such that f(x)/g(x) is decreasing on I∖{0} and g(0)=0, g(x)>0 for x>0. Then
f(x+y)(g(x)+g(y))≤g(x+y)(f(x)+f(y)), |
for x,y,x+y∈I. Moreover, if the monotoneity of f(x)/g(x) is strict, then the above inequality is also strict on I∖{0}.
The following result is a monotone form of L'Hˆopital's Rule [4] and will be useful in deriving monotoneity properties and obtaining inequalities.
Lemma 2.4. [4] For −∞<a<b<∞, let f,g:[a,b]→R be continuous on [a,b], and be differentiable on (a,b). Let g′(x)≠0 on (a,b). If f′(x)/g′(x) is increasing (decreasing) on (a,b), then so are
[f(x)−f(a)]/[g(x)−g(a)]and[f(x)−f(b)]/[g(x)−g(b)]. |
If f′(x)/g′(x) is strictly monotone, then the monotonicity in the conclusion is also strict.
The following lemma presents some known results of generalized (p,q)-elliptic integrals, which can be utilized to prove the main results of this paper.
Lemma 2.5. [14] For p,q∈(1,∞), r∈(0,1), a=1−1/p,b=a+1/q, then the functions
(1) h1(r)=Ep,q(r)−r′qKp,q(r)rq is strictly increasing and convex from (0,1) onto (aπp,q/(2b),1).
(2) h2(r)=Ep,q(r)−r′qKp,q(r)rqKp,q(r) is strictly decreasing from (0,1) onto (0,a/b).
(3) h3(r)=r′cKp,q(r) is decreasing(increasing) on (0,1) iff c≥a/b (c≤0 respectively) with h3((0,1))=(0,πp,q/2) if c≥a/b.
(4) h4(r)=Kp,q(r)−Ep,q(r)rqKp,q(r) is increasing from (0,1) onto (1/(qb),1).
(5) h5(r)=r′qKp,q(r)/Ep,q(r) is strictly decreasing from (0,1) onto itself.
Lemma 2.6. For r∈(0,1), K,p,q∈(1,∞), let s=φp,qK(r),t=φp,q1/K(r).
(1) The function f(r)=Kp,q(s)/Kp,q(r) is increasing from (0,1) onto (1,K).
(2) For q>3p−4p−1, the function g(r)=s′q/2Kp,q(s)2r′q/2Kp,q(r)2 is decreasing from (0,1) onto (0,1).
Proof. (1) According to the Lemma 2.1, we have
K2p,q(r)f′(r)=Kp,q(s)rr′qK′p,q(r){K′p,q(s)[Ep,q(s)−s′qKp,q(s)]−K′p,q(r)[Ep,q(r)−r′qKp,q(r)]}. |
Denote
f1(r)=K′p,q(r)[Ep,q(r)−r′qKp,q(r)]=rqK′p,q(r)[Ep,q(r)−r′qKp,q(r)rq]. |
By applying Lemma 2.5(1)(3) and considering s>r, we can conclude that f1(r) is increasing. Hence, we can determine that f′(r) is positive. Therefore, we can deduce that f(r) is an increasing function. For the limiting values, we have limr→0+f(r)=1 and limr→1−f(r)=K.
(2) Let
g1(r)=K′p,q(r)[qrqKp,q(r)−4(Ep,q(r)−r′qKp,q(r))]=rqK′p,q(r)Kp,q(r)[q−4Ep,q(r)−r′qKp,q(r)rqKp,q(r)]. |
By differentiation, we have
[r′q/2Kp,q(r)2]2g′(r)=−s′q/2Kp,q(s)2Kp,q(r)2rr′q/2K′p,q(r)(g1(s)−g1(r)). |
If q>3p−4p−1, g1(r) is increasing by Lemma 2.5(2) and (3), thus g′(r) is negative for s>r. Hence g(r) is decreasing, the limiting values follow from the definitions (1.2) and (1.7)
limr→0+g(r)=1,limr→1−g(r)=0. |
Lemma 2.7. For r∈(0,1) and p,q∈(1,∞), the inequality
rqKp,q(r)2N(r)>1q |
holds.
Proof. Let M(r)=rqKp,q(r)2/N(r), M1(r)=N(r)/(rqKp,q(r)), we have
M1(r)=1rqKp,q(r){qpKp,q(r)(Ep,q(r)−r′qKp,q(r))+Ep,q(r)(Kp,q(r)−Ep,q(r))−(qp−q+1)rqKp,q(r)Ep,q(r)}=qpEp,q(r)−r′qKp,q(r)rq+Ep,q(r)Kp,q(r)−Ep,q(r)rqKp,q(r)−(qp−q+1)Ep,q(r). |
According to Lemma 2.5, we have
M1(r)<qp+q(1−1p)Ep,q(r)<qπp,q2≤qKp,q(r), |
and
M(r)=Kp,q(r)/M1(r)>1/q. |
Lemma 2.8. For r∈(0,1), p,q∈(1,∞), the function
h(r)=1log(1/r)−Ep,q(r)−r′qKp,q(r)r′qKp,q(r) |
is strictly increasing from (0,1) onto (0,∞).
Proof. By differentiation, we have
h′(r)=1r(log(1/r))2−N(r)r(r′qKp,q(r))2=N(r)rq+1(log(1/r))2Kp,q(r)2[rqKp,q(r)2N(r)−(rq/2r′qlog(1r))2], |
where N(r)=qpKp,q(r)(Ep,q(r)−r′qKp,q(r))+Ep,q(r)(Kp,q(r)−Ep,q(r))−(qp−q+1)rqKp,q(r)Ep,q(r).
Let h2(r)=rq/2log(1/r), h3(r)=r′q, then h1(r)=h2(r)/h3(r), and h2(1−)=h3(1−)=0.
h′2(r)h′3(r)=−1qrq/2[q2log(1/r)−1]. |
By Lemma 2.4, the function h1(r) is strictly increasing from (0,1) onto (0,1/q). According to Lemma 2.7, we conclude that rqKp,q(r)2/N(r)>1/q. Therefore, it is easy to check that h(r) is increasing. For the limiting values, limr→0+h(r)=0. Since
limr→1−r′qKp,q(r)=0,limr→1−r′qlog(1/r)=q,limr→1−Kp,q(r)=∞, |
we have
limr→1−h(r)=limr→1−1r′qKp,q(r){r′qKp,q(r)log(1/r)−(Ep,q(r)−r′qKp,q(r))}=∞. |
In this section, we present some of the main results regarding the generalized (p,q)-elliptic integrals.
Theorem 3.1. For p,q∈(1,∞), the function Fp,q(r)=(E′p,q(r)−rqK′p,q(r))/r′q is concave on (0,r∗0) and convex on (r∗0,1) for some point r∗0∈(0,1).
Proof. Let F(r)=(Ep,q(r)−r′qKp,q(r))/rq, by the definitions (1.2) and (1.3), which can be expressed as
F(r)=Ep,q(r)−r′qKp,q(r)rq=πp,q2rq[F(−1p,1q;1−1p+1q;rq)−r′qF(1−1p,1q;1−1p+1q;rq)]=πp,q2rq[∞∑n=0((−1p,n)(1q,n)(1−1p+1q,n)n!−(1−1p,n)(1q,n)(1−1p+1q,n)n!)rqn+∞∑n=0(1−1p,n)(1q,n)(1−1p+1q,n)n!rq(n+1)]=πp,q2∞∑n=0(1−1p,n)(1q,n)(1−1p+1q,n+1)(n+1)!(1−1p)(n+1)rqn=aπp,q2(a+b)2F1(a,b;a+b+1;rq), |
with a=1−1/p,b=1/q. This implies that
Fp,q(r)=E′p,q(r)−rqK′p,q(r)r′q=aπp,q2(a+b)2F1(a,b;a+b+1;r′q). |
By differentiation, we have
F′p,q(r)=−a2bqπp,q2(a+b)(a+b+1)rq−12F1(a+1,b+1;a+b+2;r′q). |
Hence,
−2(a+b)(a+b+1)a2bqπp,qF″p,q(r)=(q−1)rq−22F1(a+1,b+1;a+b+2;r′q)−q(a+1)(b+1)a+b+2r2q−22F1(a+2,b+2;a+b+3;r′q). |
According to [4, Theorem 1.19(10)],
2F1(a,b;c;x)=(1−x)c−a−b2F1(c−a,c−b;c;x)(a+b>c,a,b,c>0), |
we have
−2(a+b)(a+b+1)a2bqπp,qF″p,q(r)=(q−1)rq−22F1(a+1,b+1;a+b+2;r′q)−q(a+1)(b+1)a+b+2rq−22F1(a+1,b+1;a+b+3;r′q)=(q−1)rq−22F1(a+1,b+1;a+b+3;r′q)[2F1(a+1,b+1;a+b+2;r′q)2F1(a+1,b+1;a+b+3;r′q)−qq−1(a+1)(b+1)a+b+2]. | (3.1) |
Let
F1(x)=2F1(a+1,b+1;a+b+2;x)2F1(a+1,b+1;a+b+3;x)=∑∞n=0anxn∑∞n=0bnxn, |
it is easy to obtain that F1(x) is strictly increasing from (0,1) onto (1,∞), since
anbn=a+b+2+na+b+2>1. |
Similarly, we can deduce that
r↦2F1(a+1,b+1;a+b+2;r′q)2F1(a+1,b+1;a+b+3;r′q)−qq−1(a+1)(b+1)a+b+2 |
is strictly decreasing from (0,1) onto (1−b(2a+b+2)(1−b)(a+b+2),∞). Therefore, the sign of F″p,q(r) changes from negative to positive on (0,1) by (3.1), we know that there exists r∗0∈(0,1) such that Fp,q(r) is concave on (0,r∗0) and convex on (r∗0,1).
Theorem 3.2. For p,q∈(1,∞), r∈(0,1), α>0, Let Hp,q(α)=αKp,q(rα), Gp,q(α)=Kp,q(rα)/α. Then
(1) The function α↦Hp,q(α) is strictly increasing and log-concave on (0,∞);
(2) The function α↦1/Hp,q(α) is strictly convex on (0,∞);
(3) The function α↦Gp,q(α) is strictly decreasing and log-convex on (0,∞).
Proof. (1) Let t=rα, then dt/dα=tlogr<0, and
dHp,q(α)dα=Kp,q(t)+αEp,q(t)−t′qKp,q(t)t′qlogr=Kp,q(t)−Ep,q(t)−t′qKp,q(t)t′qlog(1t)=Kp,q(t)log(1t)[1log(1/t)−Ep,q(t)−t′qKp,q(t)t′qKp,q(t)]. |
Hence, the monotonicity of Hp,q(α) follows from Lemma 2.8.
By logarithmic differentiation,
d(logHp,q(α))dα=1α−Ep,q(t)−t′qKp,q(t)t′qKp,q(t)log(1r)=log(1r)[1log(1/t)−Ep,q(t)−t′qKp,q(t)t′qKp,q(t)]. |
It is not difficult to verify that d(logHp,q(α))/dα is strictly increasing with respect to t by Lemma 2.8, and is strictly decreasing with respect to α. Thus the function α↦Hp,q(α) is strictly increasing and log-concave on (0,∞).
(2) Since t=rα, then α=log(1/t)/log(1/r). Differentiating 1/Hp,q(α) yields
ddα(1Hp,q(α))=−1Hp,q(α)2Kp,q(t)log(1t)[1log(1/t)−Ep,q(t)−t′qKp,q(t)t′qKp,q(t)]=−1α2Kp,q(t)2Kp,q(t)log(1t)h(t)=−(log1r)2h(t)log(1/t)Kp,q(t), | (3.2) |
where h(t) is defined in Lemma 2.8. Let f(r)=log(1/r)Kp,q(r), f1(r)=log(1/r), f2(r)=1/Kp,q(r), We clearly see that f1(1)=f2(1)=0, then
f′1(r)f′2(r)=r′qKp,q(r)2Ep,q(r)−r′qKp,q(r)=rqEp,q(r)−r′qKp,q(r)⋅r′qKp,q(r)2rq, |
which is decreasing follows from Lemma 2.5(1), (3). Hence the function f(r) is decreasing from (0,1) onto (0,∞) by Lemma 2.4. Therefore, it follows from Lemma 2.8 and (3.2) that the function α↦1/Hp,q(α) is strictly convex on (0,∞).
(3) Since t=rα, and dt/dα=tlogr<0, simple computations yields
dGp,q(α)dα=1α[Ep,q(t)−t′qKp,q(t)t′qlogr−Kp,q(t)α]=1α[Ep,q(t)−t′qKp,q(t)t′qlogr−Kp,q(t)logrlogt]=Kp,q(t)logrα[Ep,q(t)−t′qKp,q(t)t′qKp,q(t)+1log(1/t)]. |
Let g(r)=Ep,q(r)−r′qKp,q(r)r′qKp,q(r)+1log(1/r), we have
g′(r)=N(r)rq+1(log(1/r))2Kp,q(r)2[rqKp,q(r)2N(r)+(rq/2r′qlog(1r))2], |
is positive by Lemma 2.8, where the defition of N(r) is in Lemma 2.8. Hence the function g(r) is increasing from (0,1) onto (0,∞). Since logr<0, the monotonicity of the function Gp,q(α) follows immediately.
Since logGp,q(α)=logKp,q(t)−logα, by differentiation, we obtain that
d(logGp,q(α))dα=Ep,q(t)−t′qKp,q(t)t′qKp,q(t)logr−1α=Ep,q(t)−t′qKp,q(t)t′qKp,q(t)logr−logrlogt=logr[Ep,q(t)t′qKp,q(t)−1+1log(1/t)]. | (3.3) |
It follows from (3.3) and Lemma 2.5(5) that d(logGp,q(α))/dα is strictly decreasing with respect to t. Therefore, Gp,q(α) is log-convex on (0,∞) with respect to α.
Next, we apply Theorem 3.2 to obtain the inequality involving the generalized (p,q)-elliptic integrals Kp,q.
Corollary 3.1. For p,q∈(1,∞).
(1) Let α,β be positive numbers with α>β>0. The double inequality
1<Kp,q(rβ)Kp,q(rα)<αβ |
holds for all r∈(0,1).
(2) Inequality
Kp,q(√xy)≥2√log(1/x)log(1/y)log[1/(xy)]√Kp,q(x)√Kp,q(y) |
holds with equality if and only if x=y for all x,y∈(0,1).
(3) Inequality
4log[1/(xy)]Kp,q(√xy)≤1log(1/x)Kp,q(x)+1log(1/y)Kp,q(y) |
holds with equality if and only if x=y for all x,y∈(0,1).
(4) Let α,β be positive numbers with α>β>0. The double inequality
βα<Kp,q(rβ)Kp,q(rα) |
holds for all r∈(0,1).
(5) Inequality
Kp,q(√xy)≤12log(1xy)√Kp,q(x)Kp,q(y)log(1/x)log(1/y) |
holds with equality if and only if x=y for all x,y∈(0,1).
Proof. (1) By utilizing the monotonicity of the function Hp,q(α) stated in Theorem 3.2, along with the monotonicity of the function Kp,q(r), we can establish that αKp,q(rα)>βKp,q(rβ). Consequently, the double inequality holds.
(2) Since the function α↦Hp,q(α) is strictly log-concave on (0,∞), we can deduce that
logHp,q(α+β2)≥12(logHp,q(α)+logHp,q(β))⇒Hp,q(α+β2)≥√Hp,q(α)Hp,q(β) |
with equality if and only if α=β for α,β>0. For x,y∈(0,1) and set
α=log(1/x)log(1/r),β=log(1/y)log(1/r). |
Simple computations yields
Hp,q(α)=αKp,q(rα)=log(1/x)log(1/r)Kp,q(x),Hp,q(β)=βKp,q(rβ)=log(1/y)log(1/r)Kp,q(y), | (3.4) |
Hp,q(α+β2)=12log[1/(xy)]log(1/r)Kp,q(√xy). | (3.5) |
Hence, the inequality
Kp,q(√xy)≥2√log(1/x)log(1/y)log[1/(xy)]√Kp,q(x)√Kp,q(y) |
hold with equality if and only if x=y.
(3) Since the function α↦1/Hp,q(α) is strictly convex on (0,∞), we get
1Hp,q(α+β2)≤12(1Hp,q(α)+1Hp,q(β)) | (3.6) |
with equality if and only if α=β for α,β>0. Set
α=log(1/x)log(1/r),β=log(1/y)log(1/r). |
From (3.4)–(3.6), we conclude that the inequality hold with equality if and only if x=y.
(4) Since the function α↦Gp,q(α) is strictly decreasing, and the monotonicity of the function Kp,q(r), we have
Kp,q(rα)α<Kp,q(rβ)β. |
(5) Since the function α↦Gp,q(α) is log-convex on (0,∞),
logGp,q(α+β2)≤12(logGp,q(α)+logGp,q(β))⇒Gp,q(α+β2)≤√Gp,q(α)Gp,q(β) |
with equality if and only if α=β for α,β>0. Set
α=log(1/x)log(1/r),β=log(1/y)log(1/r). |
Simple computations yields
Gp,q(α)=Kp,q(rα)α=log(1/r)log(1/x)Kp,q(x),Gp,q(β)=Kp,q(rβ)β=log(1/r)log(1/y)Kp,q(y), |
Gp,q(α+β2)=2log(1/r)log[1/(xy)]Kp,q(√xy). |
Hence, the inequality
Kp,q(√xy)≤12log(1xy)√Kp,q(x)Kp,q(y)log(1/x)log(1/y) |
hold with equality if and only if x=y.
Remark 3.1. In [22], Wang et al. provided the proof for the convexity of the function (E′a−r2K′a)/r′2 and presented certain properties of the functions αKa(rα) and 1/αKa(rα) with respect to the parameter α. It is worth noting that Theorem 3.1 and Theorem 3.2(1), (2) can be reduced to [22, Theorem 1.1, Theorem 1.3] if p=q=1/a.
In this section, we study the complete monotonicity, logarithmic, geometric concavity and convexity of the generalized (p,q)-Hersch-Pfluger distortion function, and present some of the main results about φp,qK.
Theorem 4.1. For K,p,q∈(1,∞), and q>3p−4p−1, let a=1−1/p,b=1/q, f,g be defined on (0,1] by
f(r)=r−1/Kφp,qK(r),g(r)=r−Kφp,q1/K(r). |
Then f is decreaseing from (0,1] onto [1,ebR(a,b)(1−1/K)), and g is increasing from (0,1] onto (ebR(a,b)(1−K),1].
Proof. Let s=φp,qK(r), then f(r)=sr1/K, we have
(r1/K)2f′(r)=sKr1/K−1[(s′q/2Kp,q(s)r′q/2Kp,q(r))2−1]. |
Hence,
f′(r)f(r)=1Kr[(s′q/2Kp,q(s)r′q/2Kp,q(r))2−1]. |
According to Lemma 2.6(2), f′(r) is negative. Combine with f(1−)=1 and by [18, Theorem 2],
limr→0+log(r−1/Ks)=limr→0+[(μp,q(s)+logs)−1K(μp,q(r)+logr)]=bR(a,b)(1−1K). |
Let t=φp,q1/K(r), thus r=φp,qK(t) and
g(r)=φp,qK(t)−K⋅t=(t−1/Kφp,qK(t))−K=f(t)−K. |
According to the monotonicity of f(r), g(r) is increasing on (0,1]. The limiting values can also be derived from [18, Theorem 2].
Next, we utilize Theorem 4.1 to derive the inequality concerning the generalized (p,q)-Hersch-Pfluger distortion function φp,qK.
Corollary 4.1. For K,p,q∈(1,∞), and q>3p−4p−1, let a=1−1/p,b=1/q, then
(1) The double inequality
|φp,qK(r)−φp,qK(s)|≤φp,qK(|r−s|)≤ebR(a,b)(1−1/K)|r−s|1/K | (4.1) |
hold with equality if and only if r=s.
(2) The double inequality
|φp,q1/K(r)−φp,q1/K(s)|≥φp,q1/K(|r−s|)≥ebR(a,b)(1−K)|r−s|K | (4.2) |
hold with equality if and only if r=s.
Proof. (1) According to Lemma 2.3 and the monotonicity of the function f(r)=r−1/Kφp,qK(r), we can conclude that
φp,qK(x+y)≤φp,qK(x)+φp,qK(y) |
for x,y∈(0,1). Set r=x+y and s=y, we get
|φp,qK(r)−φp,qK(s)|≤φp,qK(|r−s|). |
According to f(r) is decreaseing from (0,1] onto [1,ebR(a,b)(1−1/K)), we obtain that
φp,qK(|r−s|)≤ebR(a,b)(1−1/K)|r−s|1/K |
with equality if and only if r=s.
(2) By Lemma 2.3 and the monotonicity of g(r), we have
φp,q1/K(x)+φp,q1/K(y)≤φp,q1/K(x+y) |
for x,y∈(0,1). Set r=x+y and s=y, we obtain that
|φp,q1/K(r)−φp,q1/K(s)|≥φp,q1/K(|r−s|) |
with equality if and only if r=s.
According to g(r) increasing from (0,1] onto (ebR(a,b)(1−K),1], we get
φp,q1/K(|r−s|)≥ebR(a,b)(1−K)|r−s|K |
with equality if and only if r=s. Therefore, the double inequality (4.2) hold.
Theorem 4.2. For K,p,q∈(1,∞), q>3p−4p−1, the function f(x)=log(1/φp,qK(e−x)) is increasing and convex on (0,∞), g(x)=log(1/φp,q1/K(e−x)) is increasing and concave on (0,∞), and
φp,qK(r)φp,qK(t)≤(φp,qK(√rt))2,φp,q1/K(r)φp,q1/K(t)≥(φp,q1/K(√rt))2, |
with equality if and only if K=1 for each r,t∈(0,1).
Proof. Let r=e−x,s=φp,qK(r), according to the Lemma 2.6, we have
f′(x)=1K(s′q/2Kp,q(s)r′q/2Kp,q(r))2 |
is positive and increasing with respect to x. Thus f is increasing and convex. Therefore,
f(x+y2)≤12(f(x)+f(y)), |
and putting r=e−x,t=e−y, we obtain
φp,qK(r)φp,qK(t)≤(φp,qK(√rt))2, |
with equality if and only if K=1 for each r,t∈(0,1). The proof for g(x) follows a similar approach.
Theorem 4.3. For K,p,q∈(1,∞), q>3p−4p−1, r∈(0,1), the function f(x)=φp,qK(rx)/φp,qK(x) is increasing from (0,1) onto (r1/K,φp,qK(r)), while function g(x)=φp,q1/K(rx)/φp,q1/K(x) is decreasing from (0,1) onto (φp,q1/K(r),rK). In particular,
φp,qK(rt)≤φp,qK(r)φp,qK(t),φp,q1/K(rt)≥φp,q1/K(r)φp,q1/K(t), |
with equality if and only if K=1 for each r,t∈(0,1).
Proof. Let t=rx,u=φp,qK(t),s=φp,qK(x),
f′(x)=uKsx[(u′q/2Kp,q(u)t′q/2Kp,q(t))2−(s′q/2Kp,q(s)x′q/2Kp,q(x))2], |
then
f′(x)f(x)=1Kx[(u′q/2Kp,q(u)t′q/2Kp,q(t))2−(s′q/2Kp,q(s)x′q/2Kp,q(x))2]. |
Since t<x and s′q/2Kp,q(s)x′q/2Kp,q(x) is decreasing with respect to r by Lemma 2.6, f′(x) is positive on (0,1). For the limiting values, by using L'Hˆopital's Rule, we get
limr→0+f(r)=r1/K,limr→1−h(r)=φp,qK(r). |
Since the monotonicity of the function f(x), along with the definition of the function φp,qK(r), we obtain
φp,qK(rt)≤φp,qK(r)φp,qK(t), |
with equality if and only if K=1 for each r,t∈(0,1). The proof for g(x) follows a similar approach. As a result, we will omit the detailed proof.
Theorem 4.4. For K,p∈(1,∞), r∈(0,1), q>3 and p>q(q−3)q2−2q−1, the function f(r) defined by
f(r)=arcsin(φp,qK(r))arcsin(r1/K) |
is strictly decreasing from (0,1] into [1,e(1−1/K)bR(a,b)), the function g(r) defined by
g(r)=arcsin(φp,q1/K(r))arcsin(rK) |
is strictly increasing from (0,1] into (e(1−K)bR(a,b),1], where a=1−1/p,b=a+1/q.
Proof. Let s=φp,qK(r), f1(r)=arcsin(s), f2(r)=arcsin(r1/K), f(r)=f1(r)/f2(r). Since f1(0)=f2(0)=0, we have
f′1(r)f′2(r)=sr1/K(1−r2/K1−r2)1/2s′q−1Kp,q(s)2r′q−1Kp,q(r)2. |
Let f3(r)=s′q−1Kp,q(s)2r′q−1Kp,q(r)2, according to the Lemma 2.1, we have
[r′q−1Kp,q(r)2]2f′3(r)=−s′q−1Kp,q(s)2Kp,q(r)rr′K′p,q(r)(f4(s)−f4(r)), |
where
f4(r)=K′p,q(r)[(q−1)rqKp,q(r)−2(Ep,q(r)−r′qKp,q(r))]=rqK′p,q(r)Kp,q(r)[(q−1)−2Ep,q(r)−r′qKp,q(r)rqKp,q(r)]. |
Using Lemma 2.5(2) and (3), we can observe that f4(r) is positive. Consequently, we can deduce that f3(r) is decreasing. As a result, we obtain that f(r) is decreasing on the interval (0,1] by Theorem 4.1. Furthermore, it can be deduced that the monotonicity of g(r) is similar to that of f(r).
Remark 4.1. Theorems 4.1 to 4.4 can be seen as variations and extensions of the results presented in [3, Theorem 1.14, Theorem 1.15, Theorem 6.7, Theorem 6.13]. When p=q=1/a, the results obtained in Theorems 4.1 to 4.4 can be reduced to those obtained in [3].
In this paper, we investigate the properties of the generalized (p,q)-elliptic integrals and the generalized (p,q)-Hersch-Pfluger distortion function. Through our analysis, we have established complete monotonicity, logarithmic, geometric concavity, and convexity properties for certain functions involving these integrals and arcsine functions. These properties provide valuable insights into the behavior of these functions. Furthermore, we have derived several sharp functional inequalities for the generalized (p,q)-elliptic integrals and the generalized (p,q)-Hersch-Pfluger distortion function. These inequalities not only improve upon existing results but also generalize them.
The authors declare that they have not used Artificial Intelligence tools in the creation of this article.
The research was supported by the Natural Science Foundation of China (Grant Nos. 11601485, 11401531).
The authors declare that they have no conflicts of interest.
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