The combat domain of modern warfare is becoming increasingly multidimensional. It is important to evaluate the resilience of the air-ground cooperative network for defending against attack threats and recovery performance. First, a resilience analysis model was proposed to effectively analyze and evaluate the resilience of the air-ground cooperative network. Then, considering the available resources, three dynamic reconfiguration strategies were given from the global perspective to help the air-ground cooperative network quickly recover performance and enhance combat capabilities. Finally, a typical 50-node network was taken as an example to prove the effectiveness and feasibility of the proposed model. The proposed method can provide scientific guidance for improving the air-ground cooperative network combat capabilities.
Citation: Xiaoyang Xie, Shanghua Wen, Minglong Li, Yong Yang, Songru Zhang, Zhiwei Chen, Xiaoke Zhang, Hongyan Dui. Resilience evaluation and optimization for an air-ground cooperative network[J]. Electronic Research Archive, 2024, 32(5): 3316-3333. doi: 10.3934/era.2024153
[1] | M. Emin Özdemir, Saad I. Butt, Bahtiyar Bayraktar, Jamshed Nasir . Several integral inequalities for (α, s,m)-convex functions. AIMS Mathematics, 2020, 5(4): 3906-3921. doi: 10.3934/math.2020253 |
[2] | Tekin Toplu, Mahir Kadakal, İmdat İşcan . On n-Polynomial convexity and some related inequalities. AIMS Mathematics, 2020, 5(2): 1304-1318. doi: 10.3934/math.2020089 |
[3] | Haoliang Fu, Muhammad Shoaib Saleem, Waqas Nazeer, Mamoona Ghafoor, Peigen Li . On Hermite-Hadamard type inequalities for $ n $-polynomial convex stochastic processes. AIMS Mathematics, 2021, 6(6): 6322-6339. doi: 10.3934/math.2021371 |
[4] | Muhammad Samraiz, Kanwal Saeed, Saima Naheed, Gauhar Rahman, Kamsing Nonlaopon . On inequalities of Hermite-Hadamard type via $ n $-polynomial exponential type $ s $-convex functions. AIMS Mathematics, 2022, 7(8): 14282-14298. doi: 10.3934/math.2022787 |
[5] | Shuang-Shuang Zhou, Saima Rashid, Muhammad Aslam Noor, Khalida Inayat Noor, Farhat Safdar, Yu-Ming Chu . New Hermite-Hadamard type inequalities for exponentially convex functions and applications. AIMS Mathematics, 2020, 5(6): 6874-6901. doi: 10.3934/math.2020441 |
[6] | Hong Yang, Shahid Qaisar, Arslan Munir, Muhammad Naeem . New inequalities via Caputo-Fabrizio integral operator with applications. AIMS Mathematics, 2023, 8(8): 19391-19412. doi: 10.3934/math.2023989 |
[7] | Saad Ihsan Butt, Ahmet Ocak Akdemir, Muhammad Nadeem, Nabil Mlaiki, İşcan İmdat, Thabet Abdeljawad . $ (m, n) $-Harmonically polynomial convex functions and some Hadamard type inequalities on the co-ordinates. AIMS Mathematics, 2021, 6(5): 4677-4690. doi: 10.3934/math.2021275 |
[8] | Ahmet Ocak Akdemir, Saad Ihsan Butt, Muhammad Nadeem, Maria Alessandra Ragusa . Some new integral inequalities for a general variant of polynomial convex functions. AIMS Mathematics, 2022, 7(12): 20461-20489. doi: 10.3934/math.20221121 |
[9] | Gültekin Tınaztepe, Sevda Sezer, Zeynep Eken, Sinem Sezer Evcan . The Ostrowski inequality for $ s $-convex functions in the third sense. AIMS Mathematics, 2022, 7(4): 5605-5615. doi: 10.3934/math.2022310 |
[10] | Moquddsa Zahra, Dina Abuzaid, Ghulam Farid, Kamsing Nonlaopon . On Hadamard inequalities for refined convex functions via strictly monotone functions. AIMS Mathematics, 2022, 7(11): 20043-20057. doi: 10.3934/math.20221096 |
The combat domain of modern warfare is becoming increasingly multidimensional. It is important to evaluate the resilience of the air-ground cooperative network for defending against attack threats and recovery performance. First, a resilience analysis model was proposed to effectively analyze and evaluate the resilience of the air-ground cooperative network. Then, considering the available resources, three dynamic reconfiguration strategies were given from the global perspective to help the air-ground cooperative network quickly recover performance and enhance combat capabilities. Finally, a typical 50-node network was taken as an example to prove the effectiveness and feasibility of the proposed model. The proposed method can provide scientific guidance for improving the air-ground cooperative network combat capabilities.
In recent years, convexity theory has gained special attention by many researchers because of it engrossing properties and expedient characterizations. It has many applications in fields like biology, numerical analysis and statistics (see [1,2,3,4]). Mathematical inequalities are extensively studied with all type of convex functions (see[1,3,11,13,14,16]). One of the fundamental inequality is Hermite-Hadamard inequality. It has been discussed via different types of convexities and became the center of attention for many researchers. Recently, in 2016, Khan et al. have discussed generalizations of Hermite-Hadamard type for MT-convex functions [26]. In 2017, Khan et al. studied some new inequalities of Hermite-Hadamard types [27]. In 2019, Khurshid et al. have utilized conformable fractional integrals via preinvex functions [28]. In 2020, Khan et al. have discussed Hermite-Hadamard type inequalities via quantum calculus involving green function [29], Mohammed et al. have established a new version of Hermite-Hadamard inequality for Riemann-Liouville fractional integrals [30], Han et al. used fractional integral to generalize Hermite-Hadamard inequality for convex functions [31], Zhao et al. utilized harmonically convex functions to generalized fractional integral inequalities of Hermite-Hdamrd type [32], Awan et al. presented new inequalities of Hermite-Hdamard type for n-polynomial harmonically convex functions [33]. In 2022, Khan et al. introduced some new versions of Hermite-Hadamard integral inequalities in fuzzy fractional calculus for generalized pre-invex functions via fuzzy-interval-valued settings [34]. This reflects the importance of Hermite Hadamard type inequalities among current research.
In [9], s-convex function is given as,
Definition 1.1. A real valued function χ is called s-convex function on R, if
χ(ςρ+(1−ς)γ)≤ςsχ(ρ)+(1−ς)sχ(γ), |
for each ρ,γ∈R and ς∈(0,1) where s∈(0,1].
In [10], m-convexity is discussed as,
Definition 1.2. A real valued function χ defined on [0,b] is said to be a m-convex function for m∈[0,1], if
χ(ςρ+m(1−ς)γ)≤ςχ(ρ)+m(1−ς)χ(γ), |
holds for all ρ,γ∈[0,b] and ς∈[0,1].
(s,m)-convexity in [17] is discussed as,
Definition 1.3. A function χ:[0,b]⟶R, b>0 is said to be a (s,m)-convex function in the second sense where s,m∈(0,1]2, if
χ(ςρ+m(1−ς)γ)≤ςsχ(ρ)+m(1−ς)sχ(γ), |
holds provided that all ρ,γ∈[0,b] and ς∈[0,1].
Equivalent definition for (s,m)–convex functions:
Let ρ,α,γ∈[0,b], ρ<α<γ
χ(α)≤(γ−αγ−ρ)sχ(ρ)+m(α−ργ−ρ)sχ(γ). | (1.1) |
Hölder-İşcan Inequality [5]:
Let p>1, χ and ψ be real valued functions defined on [ρ,γ] and |χ|p,|ψ|q are integrable functions on interval [ρ,γ]
∫γρ|χ(ω)ψ(ω)|dω≤1γ−ρ(∫γρ(γ−ω)|χ(ω)|pdω)1p(∫γρ(γ−ω)|ψ(ω)|qdω)1q+1γ−ρ(∫γρ(ω−ρ)|χ(ω)|pdω)1p(∫γρ(ω−ρ)|ψ(ω)|qdω)1q, | (1.2) |
where 1p+1q=1.
Following lemma is useful to obtain our main results.
Lemma 1.4. [8] For n∈N, let χ:U⊆R⟶R be n-times differentiable mapping on U∘, where ρ,γ∈U∘, ρ<γ and χn∈L[ρ,γ], we have following identity
n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−γ∫ρχ(ω)dω=(−1)n+1n!γ∫ρωnχ(n)(ω)dω, | (1.3) |
where an empty set is understood to be nil.
In this paper, Hölder-İşcan inequality is used to modify inequalities involving functions having s-convex or s-concave derivatives at certain powers. The purpose of this paper is to establish some generalized inequalities for n-times differentiable (s,m)-convex functions. Applications of these inequalities to means are also discussed. Means are defined as,
Let 0<ρ<γ,
A(ρ,γ)=ρ+γ2, |
G(ρ,γ)=√ργ, |
Lp(ρ,γ)=(γp+1−ρp+1(p+1)(γ−ρ))1p, |
where p≠0,−1 and ρ≠γ.
Theorem 2.1. For any positive integer n, let χ:U⊆(0,∞)→R be n-times differentiable mapping on U∘, where ρ,γ∈U∘ with ρ<γ. If χ(n)∈L[ρ,γ] and |χ(n)|q for q>1 is (s,m)-convex on interval [ρ,γ] then
|n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−μ∫ρχ(ω)dω|≤1n!(γ−ρ)1q([γLnpnp(ρ,γ)−Lnp+1np+1(ρ,γ)]1p[|χn(γ)|q(s+2)(s+1)+m|χn(ρ)|q(s+2)]1q+[Lnp+1np+1(ρ,γ)−ρLnpnp(ρ,γ)]1p[|χn(γ)|q(s+2)+m|χn(ρ)|q(s+1)(s+2)]1q), | (2.1) |
where 1p+1q=1.
Proof. Since |χn|q is (s,m)-convex by using inequality (1.1) for ρ<ω<γ, using Lemma 1.4 and Hölder-Işcan inequality (1.2),
|χn(ω)|q≤|χn(ω−ργ−ργ+mγ−ωγ−ρρ)|q≤(ω−ργ−ρ)s|χn(γ)|q+m(γ−ωγ−ρ)s|χn(ρ)|q,|n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−γ∫ρχ(ω)dω|≤1n!γ∫ρωn|χ(n)(ω)|dω,≤1n!1γ−ρ{(γ∫ρ(γ−ω)ωnpdω)1p(γ∫ρ(γ−ω)|χ(n)(ω)|qdω)1q+(γ∫ρ(ω−ρ)ωnpdω)1p(γ∫ρ(ω−ρ)|χ(n)(ω)|qdω)1q},≤1n!1γ−ρ(γ∫ρ(γ−ω)ωnpdω)1p(γ∫ρ(γ−ω)[(ω−ργ−ρ)s|χn(γ)|q+m(γ−ωγ−ρ)s|χn(ρ)|q]dω)1q+1n!1γ−ρ(γ∫ρ(ω−ρ)ωnpdω)1p(γ∫ρ(ω−ρ)[(ω−ργ−ρ)s|χn(γ)|q+m(γ−ωγ−ρ)s|χn(ρ)|q]dω)1q, | (2.2) |
Let
I1=[γ∫ρ(γ−ω)ωnpdω]1p=[γ∫ρ(γωnp−ωnp+1)dω]1p=(γ−ρ)1p[γ(γnp+1−ρnp+1(γ−ρ)(np+1))−(γnp+2−ρnp+2(γ−ρ)(np+2))]1p=(γ−ρ)1p[γLnpnp(ρ,γ)−Lnp+1np+1(ρ,γ)]1p, |
I2=[γ∫ρ(ω−ρ)ωnpdt]1p=[γ∫ρ(ωnp+1−ρωnp)dω]1p=(γ−ρ)1p[(γnp+2−ρnp+2(γ−ρ)(np+2))−ρ(γnp+1−ρnp+1(γ−ρ)(np+1))]1p=(γ−ρ)1p[Lnp+1np+1(ρ,γ)−ρLnpnp(ρ,γ)]1p, |
I3=γ∫ρ(γ−ω)(ω−ρ)sdω=(γ−ω)(ω−ρ)s+1s+1|γρ+γ∫ρ(ω−ρ)s+1s+1dω=(γ−ρ)s+2(s+1)(s+2), |
I4=γ∫ρ(γ−ω)s+1dω=(γ−ρ)s+2s+2,I5=γ∫ρ(ω−ρ)s+1dω=(γ−ρ)s+2s+2,I6=γ∫ρ(ω−ρ)(γ−ω)sdω=(ω−ρ)(γ−ω)s+1(s+1)|γρ+γ∫ρ(γ−ω)s+1(s+1)dω=(γ−ρ)s+2(s+1)(s+2). |
Substituting integrals I1,I2,I3,I4,I5,I6 in inequality (2.2) we have,
|n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−γ∫ρχ(ω)dω|≤1n!(γ−ρ)((γ−ρ)1p[γLnpnp(ρ,γ)−Lnp+1np+1(ρ,γ)]1p[(γ−ρ)2(|χn(γ)|q(s+2)(s+1)+m|χn(ρ)|q(s+2))]1q+(γ−ρ)1p[Lnp+1np+1(ρ,γ)−ρLnpnp(ρ,γ)]1p[(γ−ρ)2(|χn(γ)|q(s+2)+m|χn(ρ)|q(s+1)(s+2))]1q) |
=(γ−ρ)1p−1+2qn!([γLnpnp(ρ,γ)−Lnp+1np+1(ρ,γ)]1p[|χn(γ)|q(s+2)(s+1)+m|χn(ρ)|q(s+2)]1q+[Lnp+1np+1(ρ,γ)−ρLnpnp(ρ,γ)]1p[|χn(γ)|q(s+2)+m|χn(ρ)|q(s+1)(s+2)]1q) |
=1n!(γ−ρ)1q([γLnpnp(ρ,γ)−Lnp+1np+1(ρ,γ)]1p[|χn(γ)|q(s+2)(s+1)+m|χn(ρ)|q(s+2)]1q+[Lnp+1np+1(ρ,γ)−ρLnpnp(ρ,γ)]1p[|χn(γ)|q(s+2)+m|χn(ρ)|q(s+1)(s+2)]1q). |
which is required inequality (2.1).
For n=1 inequality (2.1) becomes,
|(χ(γ)γ−χ(ρ)ργ−ρ)−1γ−ργ∫ρχ(ω)dω|≤(γ−ρ)1q−1([γLpp(ρ,γ)−Lp+1p+1(ρ,γ)]1p[|χ′(γ)|q(s+1)(s+2)+m|χ′(ρ)|(s+2)q]1q+[Lp+1p+1(ρ,γ)−ρLpp(ρ,γ)]1p[m|χ′(ρ)|q(s+1)(s+2)+|χ′(γ)|(s+2)q]1q). | (2.3) |
Remark 2.2. For s=1 and m=1 our resulting inequality (2.1) becomes the inequality (2) of [5].
Theorem 2.3. For n∈N, let χ:U⊆(0,∞)→R be n-times differentiable mapping on U∘, where, ρ,γ∈U∘, ρ<γ, χ(n)∈L[ρ,γ] and |χ(n)|q for q>1, is (s,m)-convex on interval [ρ,γ] then following inequality holds
|n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−γ∫ρχ(ω)dω|≤1s1qn!(12)1p(γ−ρ)2p−1((|χ(n)(γ)|q(γ−ρ)s−1[−Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)−ργLnqnq(ρ,γ)]+m|χ(n)(ρ)|q(γ−ρ)s−1[Lnq+2nq+2(ρ,γ)−2γLnq+1nq+1(ρ,γ)+γ2Lnqnq(ρ,γ)])1q+(|χ(n)(γ)|q(γ−ρ)s−1[Lnq+2nq+2(ρ,γ)−2ρLnq+1nq+1(ρ,γ)+ρ2Lnqnq(ρ,γ)]+m|χ(n)(ρ)|q(γ−ρ)s−1[−Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)−ργLnqnq(ρ,γ)])1q). | (2.4) |
Proof. Since |χ(n)|q for q>1 is (s,m)-convex on [ρ,γ], by using Lemma 1.4 and Hölder-İşcan inequality (1.2), since s∈(0,1], this fact can be used for ω,ρ,γ∈U⊆(0,∞),
(ω−ρ)s<(ω−ρ)s,(γ−ω)s<(γ−ω)s|n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−γ∫ρχ(ω)dω|≤1n!γ∫ρ1.ωn|χ(n)(ω)|dω,≤1n!1(γ−ρ)([(γ∫ρ(γ−ω)dω)1p(γ∫ρ(γ−ω)ωnq|χ(n)(ω)|qdω)1q]+[(γ∫ρ(ω−ρ)dω)1p(γ∫ρ(ω−ρ)ωnq|χ(n)(ω)|qdω)1q]),≤1n!1(γ−ρ)(γ∫ρ(γ−ω)dω)1p(γ∫ρ(γ−ω)ωnq[(ω−ργ−ρ)s|χn(γ)|q+m(γ−ωγ−ρ)s|χn(ρ)|q]dt)1q+1n!1(γ−ρ)(γ∫ρ(ω−ρ)dt)1p(γ∫ρ(ω−ρ)ωnq[(ω−ργ−ρ)s|χn(γ)|q+m(γ−ωγ−ρ)s|χn(ρ)|q]dx)1q,≤1s1qn!1(γ−ρ)(γ∫ρ(γ−ω)dω)1p(γ∫ρ(γ−ω)ωnq[(ω−ρ)(γ−ρ)s|χn(γ)|q+m(γ−ω)(γ−ρ)s|χn(ρ)|q]dω)1q+1s1qn!1(γ−ρ)(γ∫ρ(ω−ρ)dω)1p(γ∫ρ(ω−ρ)ωnq[(ω−ρ)(γ−ρ)s|χn(γ)|q+m(γ−ω)(γ−ρ)s|χn(ρ)|q]dω)1q, I1=γ∫ρ(γ−ω)dω=(γ−ρ)22 I2=γ∫ρ(γ−ω)(ω−ρ)ωnqdω=γωnq+1nq+1−ργωnq+1nq+1−ωnq+3nq+3+ρωnq+2nq+2|γρ =−(γnq+3−ρnq+3nq+3)+ρ(γnq+2−ρnq+2nq+2)+γ(γnq+2−ρnq+2nq+2)−ργ(γnq+1−ρnq+1nq+1) =(γ−ρ)[−Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)−ργLnqnq(ρ,γ)], I3=γ∫ρ(γ−ω)2ωnqdω=γ2ωnq+1nq+1+ωnq+3nq+3−2γωnq+2nq+2|γρ =(γnq+3−ρnq+3nq+3)−2γ(γnq+2−ρnq+2nq+2)+γ2(γnq+1−ρnq+1nq+1) =(γ−ρ)[Lnq+2nq+2(ρ,γ)−2γLnq+1nq+1(ρ,γ)+γ2Lnqnq(ρ,γ)], I4=γ∫ρ(ω−ρ)2ωnqdω=ωnq+3nq+3+ρ2ωnq+1nq+1−2ρωnq+2nq+2|γρ =(γnq+3−ρnq+3nq+3)+ρ2(γnq+1−ρnq+1nq+1)−2ρ(γnq+2−ρnq+2nq+2) =(γ−ρ)[Lnq+2nq+2(ρ,γ)+ρ2Lnqnq(ρ,γ)−2ρLnq+1nq+1(ρ,γ)]. | (2.5) |
Substituting integrals I1,I2,I3,I4,I5,I6 in inequality (2.5) we have,
|n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−γ∫ρχ(ω)dω|≤1s1qn!(12)1p(γ−ρ)2p−1×((|χ(n)(γ)|q(γ−ρ)s[(γ−ρ)(−Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)−ργLnqnq(ρ,γ))]+m|χ(n)(ρ)|q(γ−ρ)s[(γ−ρ)(Lnq+2nq+2(ρ,γ)−2γLnq+1nq+1(ρ,γ)+γ2Lnqnq(ρ,γ))])1q+(|χ(n)(γ)|q(γ−ρ)s[(γ−ρ)(Lnq+2nq+2(ρ,γ)−2ρLnq+1nq+1(ρ,γ)+ρ2Lnqnq(ρ,γ))]+m|χ(n)(ρ)|q(γ−ρ)s[(γ−ρ)(−Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)−ργLnqnq(ρ,γ))])1q), |
=1s1qn!(12)1p(γ−ρ)2p−1×((|χ(n)(γ)|q(γ−ρ)s−1[−Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)−ργLnqnq(ρ,γ)]+m|χ(n)(ρ)|q(γ−ρ)s−1[Lnq+2nq+2(ρ,γ)−2γLnq+1nq+1(ρ,γ)+γ2Lnqnq(ρ,γ)])1q+(|χ(n)(γ)|q(γ−ρ)s−1[Lnq+2nq+2(ρ,γ)−2ρLnq+1nq+1(ρ,γ)+ρ2Lnqnq(ρ,γ)]+m|χ(n)(ρ)|q(γ−ρ)s−1[−Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)−ργLnqnq(ρ,γ)])1q). |
For n=1, Theorem2.3 reduced to the inequality
|γχ(γ)−ρχ(ρ)(γ−ρ)−1(γ−ρ)γ∫ρχ(ω)dω|≤1s1q(12)1p(γ−ρ)2p−2((|χ(1)(γ)|q(γ−ρ)s−1[−Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)−ργLqq(ρ,γ)]+m|χ(1)(ρ)|(γ−ρ)s−1q[Lq+2q+2(ρ,γ)−2γLq+1q+1(ρ,γ)+γ2Lqq(ρ,γ)])1q+(|χ(1)(γ)|(γ−ρ)s−1q[Lq+2q+2(ρ,γ)−2ρLq+1q+1(ρ,γ)+ρ2Lqq(ρ,γ)]+m|χ(1)(ρ)|(γ−ρ)s−1q[−Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)−ργLqq(ρ,γ)])1q). | (2.6) |
Remark 2.4. For s=1 and m=1 our resulting inequality (2.4) becomes the inequality (6) of [5].
Theorem 2.5. If function χ:[0,b]⟶R, b>0 is a (s, m)-convex function in the second sense where (s,m)∈(0,1]2, holds provided that all ρ,γ∈[0,b] and ς∈[0,1], then
2sχ(ρ+mγ2)≤[1mγ−ρmγ∫ρχ(ω)dω+m2mγ−ργ∫ρmχ(l)dl]≤χ(ρ)+mχ(γ)s+1+χ(γ)+mχ(ρm2)s+1. | (2.7) |
Proof. A function χ:[0,b]⟶R, b>0 is said to be a (s,m)-convex function in the second sense where s,m∈(0,1]2, if
χ(ςρ+m(1−ς)γ)≤ςsχ(ρ)+m(1−ς)sχ(γ), |
holds provided that all ρ,γ∈[0,b] and ς∈[0,1].
Integrating w.r.t ς on [0,1],
1∫0χ(ςρ+m(1−ς)γ)dς≤1∫0ςsχ(ρ)dς+1∫0m(1−ς)sχ(γ)dς,=ςs+1s+1|10χ(ρ)−mχ(γ)(1−ς)s+1s+1|10=χ(ρ)+mχ(γ)s+1. 1∫0χ(ςρ+m(1−ς)γ)dς≤χ(ρ)+mχ(γ)s+1. | (2.8) |
and
χ(ςγ+m(1−ς)ρm2)≤ςsχ(γ)+m(1−ς)sχ(ρm2),1∫0χ(ςγ+m(1−ς)ρm2)dς≤χ(γ)+mχ(ρm2)s+1. | (2.9) |
As χ is (s,m)-convex,
χ(ρ+mγ2)=χ(ςρ+(1−ς)mγ2+m.(1−ς)ρm+ςγ2)≤(12)sχ(ςρ+(1−ς)γm)+m(12)sχ(ςγ+(1−ς)ρm), |
Integrating w.r.t ς over [0,1] and by using (2.8) and (2.9) we get,
2sχ(ρ+mγ2)≤1∫0(χ(ςρ+(1−ς)γm)dς+m1∫0χ(ςγ+(1−ς)ρm)dς≤χ(ρ)+mχ(γ)s+1+χ(γ)+mχ(ρm2)s+1. | (2.10) |
Substituting in first integral,
ςρ+(1−ς)γm=ω,
1∫0χ(ςρ+(1−ς)mγ)dς=1γm−ργm∫ρχ(ω)dω. | (2.11) |
Substituting in the second integral,
ςγ+(1−ς)ρm=l,
1∫0χ(ςγ+(1−ς)ρm)dς=mγm−ργ∫ρmχ(l)dl, | (2.12) |
Using (2.11) and (2.12) in (2.10) required inequality (2.7) obtained.
Remark 2.6. For s,m=1 inequality (2.7) becomes classical Hadamard inequality for convex functions.
Theorem 2.7. For n∈N, let χ:U⊆(0,∞)→R be n-times differentiable mapping on U∘, where, ρ,γ∈U∘, ρ<γ and χ(n)∈L[ρ,γ] and |χ(n)|q for q>1 is (s, m)-concave on interval [ρ,mγ], then
|n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−mγ∫ρχ(ω)dω|≤2sq(mγ−ρ)1q|χ(n)(ρ+mγ2)|n!((γLnpnp(ρ,mγ)−Lnp+1np+1(ρ,mγ))1p+(Lnp+1np+1(ρ,mγ)−ρLnpnp(ρ,mγ))1p). | (2.13) |
Proof. |χ(n)|q for q>1 is (s,m)-concave then by using Theorem 2.5 we have,
|χ(n)(ρ)|q+m|χ(n)(γ)|qs+1+|χ(n)(γ)|q+m|χ(n)(ρm2)|qs+1−m2(mγ−ρ)γ∫ρm|χ(n)(l)|qdl≤1(mγ−ρ)mγ∫ρ|χ(n)(ω)|qdω≤2s|χ(n)(ρ+mγ2)|q, |
mγ∫ρ|χ(n)(ω)|qdω≤2s(mγ−ρ)|χ(n)(ρ+mγ2)|q, |
1(mγ−ρ)γm∫ρ(γ−ω)|χ(n)(ω)|qdω≤γm∫ρ|χ(n)(ω)|qdω≤2s(mγ−ρ)|χ(n)(ρ+mγ2)|q, |
1(mγ−ρ)γm∫ρ(γ−ω)|χ(n)(ω)|qdω≤γm∫ρ|χ(n)(ω)|qdω≤2s(mγ−ρ)|χ(n)(ρ+mγ2)|q. |
Using Lemma 1.4 and Hölder-Îşcan inequality (1.2),
|n−1∑ν=0(−1)ν(χ(ν)(γ)γν+1−χ(ν)(ρ)ρν+1(ν+1)!)−γm∫ρχ(ω)dω|≤1n!γm∫ρωn|χ(n)(ω)|dω, ≤1n!1γ−ρ{(γm∫ρ(γ−ω)ωnpdω)1p(mγ∫ρ(γ−ω)|χn(ω)|qdω)1q+(γm∫ρ(ω−ρ)ωnpdω)1p(mγ∫ρ(ω−ρ)|χn(ω)|qdω)1q}, ≤1n!1γ−ρ((γm∫ρ(γ−ω)ωnpdω)1p(2s(mγ−ρ)2|χ(n)(ρ+mγ2)|q)1q+(γm∫ρ(ω−ρ)ωnpdω)1p(2s(mγ−ρ)2|χ(n)(ρ+mγ2)|q)1q), I1=(γm∫ρ(γ−ω)ωnpdω)1p=(γωnp+1np+1|γmρ−ωnp+2np+2|γmρ)1p =(mγ−ρ)1p(γLnpnp(ρ,mγ)−Lnp+1np+1(ρ,mγ))1p, I2=(γm∫ρ(ω−ρ)ωnpdω)1p=(ωnp+2np+2|γmρ−ρωnp+1np+1|γmρ)1p =(mγ−ρ)1p(Lnp+1np+1(ρ,mγ)−ρLnpnp(ρ,mγ))1p. | (2.14) |
Substituting integrals I1,I2 in inequality (2.14) required inequality (2.13) is obtained.
For n=1 inequality (2.13) becomes,
|χ(γ)γ−ρχ(ρ)(γ−ρ)−1(γ−ρ)γm∫ρχ(ω)dω|≤2sq(mγ−ρ)1q|χ(1)(ρ+γ2)|1!((γLpp(ρ,mγ)−Lp+1p+1(ρ,mγ))1p+(Lp+1p+1(ρ,mγ)−ρLpp(ρ,mγ))1p). | (2.15) |
Remark 2.8. For s=1 and m=1 our resulting inequality becomes the inequality obtained in Theorem 4 of [5].
Proposition 2.9. Let ρ,γ∈(0,∞), where ρ<γ, q>1, n,i∈N with i≥n,
|Lii(ρ,γ)[(i+1)∑n−1ν=0(−1)νP(i,ν)(ν+1)!−1]|≤1n!(γ−ρ)1q−1×([γLnpnp(ρ,γ)−Lnp+1np+1(ρ,γ)]1p(γ(i−n)q(s+1)(s+2)+mρ(i−n)q(s+2))1q+[Lnp+1np+1(ρ,γ)−ρLnpnp(ρ,γ)]1p(mρ(i−n)q(s+1)(s+2)+γ(i−n)q(s+2))1q), | (2.16) |
where
P(i,n)={i(i−1)...(i−n+1),i>nn!,i=n1,n=0}. |
Proof. Let
χ(ω)=ωi,|χ(n)(ω)|q=|P(i,n)ωi−n|q |
Let
g(ς)=|P(i,n)(ςρ+m(1−ς)γ|(i−n)q−|P(i,n)ςsρ|(i−n)q−|mP(i,n)(1−ς)sγ|(i−n)q, |
g″ |
means g is convex and , which omplies , hence
By using Theorem 2.1 for which is –convex for inequality (2.16) obtained.
Remark 2.10. For inequality (2.16) becomes inequality of [5].
Example 2.11. Taking , , in Proposition 2.9, the following is valid:
where and are classical arithmetic and geometric means, respectively.
Proposition 2.12. Let , with, , and ,
(2.17) |
where is classical logarithmic mean.
Proof.
Let
means is convex and which implies as
So is -convex. Then by using inequality (2.3) required inequality (2.17) obtained.
Remark 2.13. For inequality (2.17) becomes of [5].
Example 2.14. For and , Proposition 2.12 gives:
Proposition 2.15. Let , , ,
then
(2.18) |
Proof.
Let
and so and is -convex, by using inequality (2.3) we have (2.18).
Remark 2.16. For inequality (2.18) becomes of [5].
Example 2.17. For and Proposition 2.15 reduced to
(2.19) |
Proposition 2.18. Let with , and then we have
(2.20) |
Proof. Let,
As is -convex on , therefore by using Theorem 2.3 required inequality (2.20) is obtained.
Remark 2.19. For inequality (2.20) becomes inequality obtained in Proposition of [5].
Proposition 2.20. Let with and then we have,
(2.21) |
Proof.
As is –convex, therefore by using inequality (2.6) required (2.21) obtained.
Remark 2.21. For inequality (2.21) becomes inequality obtained in Proposition of [5].
Proposition 2.22. Let with and , then
(2.22) |
Proof.
is -convex by using inequality (2.6) required (2.22) obtained.
For inequality (2.22) becomes,
(2.23) |
Remark 2.23. For inequality (2.22) becomes inequality obtained in Proposition of [5].
Proposition 2.24. Let with , and we have,
(2.24) |
Proof.
As is -concave by using inequality (2.15) we obtain required inequality (2.24).
Remark 2.25. For inequality (2.24) becomes the inequality obtained in Proposition of [5].
In this paper, Hölder-Isçan inequality is utilized to prove Hermite-Hadamard type inequalities for -times differentiable -convex functions. The method is adequate and provide many generalizations of existing results as shown in remarks. Moreover, many other inequalities can be generalized for other types of convex functions.
This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation, (grant number B05F650018)
The authors declare no conflict of interest.
[1] |
Y. Sun, Z. Fang, Research on projection gray target model based on FANP-QFD for weapon system of systems capability evaluation, IEEE Syst. J., 15 (2020), 4126–4136. https://doi.org/10.1109/JSYST.2020.3027585 doi: 10.1109/JSYST.2020.3027585
![]() |
[2] |
X. Wang, Y. Zhang, L. Wang, D. Lu, G. Zeng, Robustness evaluation method for unmanned aerial vehicle swarms based on complex network theory, Chin. J. Aeronaut., 33 (2020), 352–364. https://doi.org/10.1016/j.cja.2019.04.025 doi: 10.1016/j.cja.2019.04.025
![]() |
[3] |
P. Uday, R. Chandrahasa, K. Marais, System importance measures: Definitions and application to system-of-systems analysis, Reliab. Eng. Syst. Saf., 191 (2019), 106582. https://doi.org/10.1016/j.ress.2019.106582 doi: 10.1016/j.ress.2019.106582
![]() |
[4] |
Z. Chen, Z. Zhou, L. Zhang, C. Cui, J. Zhong, Mission reliability modeling and evaluation for reconfigurable unmanned weapon system-of-systems based on effective operation loop, J. Syst. Eng. Electron., 34 (2023), 588–597. https://doi.org/10.23919/JSEE.2023.000082 doi: 10.23919/JSEE.2023.000082
![]() |
[5] |
K. Yang, J. Li, M. Liu, Complex systems and network science: a survey, J. Syst. Eng. Electron., 34 (2023), 543–573. https://doi.org/10.23919/JSEE.2023.000080 doi: 10.23919/JSEE.2023.000080
![]() |
[6] |
J. Sun, B. Ge, J. Li, K. Yang, Operation network modeling with degenerate causal strengths for missile defense systems, IEEE Syst. J., 12 (2016), 274–284. https://doi.org/10.1109/JSYST.2016.2570519 doi: 10.1109/JSYST.2016.2570519
![]() |
[7] | J. R. Cares, R. J. Christian, R. C. Manke, Fundamentals of distributed, networked military forces and the engineering of distributed systems, NUWC-NPT Tech. Rep., 11 (2002), 200–209. https://www.researchgate.net/profile/Jeff-Cares/publication/235107120 |
[8] |
J. Li, B. Ge, K. Yang, Y. Chen, Y. Tan, Meta-path based heterogeneous combat network link prediction, Phys. A: Stat. Mech. Appl., 482 (2017), 507–523. https://doi.org/10.1016/j.physa.2017.04.126 doi: 10.1016/j.physa.2017.04.126
![]() |
[9] | J. Li, D. Zhao, J. Jiang, K. Yang, Y. Chen, Capability oriented equipment contribution analysis in temporal combat networks, IEEE Trans. Syst. Man Cybern.: Syst., 51 (2018), 696–704. https://doi.org/0.1109/TSMC.2018.2882782 |
[10] |
J. Li, J. Jiang, K. Yang, Y. Chen, Research on functional robustness of heterogeneous combat networks, IEEE Syst. J., 13 (2018), 1487–1495. https://doi.org/10.1109/JSYST.2018.2828779 doi: 10.1109/JSYST.2018.2828779
![]() |
[11] |
J. Sun, J. Li, Y. You, J. Jiang, B. Ge, Combat network link prediction based on embedding learning, J. Syst. Eng. Electron., 33 (2022), 345–353. https://doi.org/10.23919/JSEE.2022.000036 doi: 10.23919/JSEE.2022.000036
![]() |
[12] | L. Chen, C. Wang, C. Zeng, L. Wang, H. Liu, J. Chen, A novel method of heterogeneous combat network disintegration based on deep reinforcement learning, Front. Phys., 10 (2022), https://doi.org/1021245.10.3389/fphy.2022.1021245 |
[13] |
C. Cheng, G. Bai, Y. Zhang, J. Tao, Resilience evaluation for UAV swarm performing joint reconnaissance mission, Chaos, 29 (2019), 190–200. https://doi.org/10.1063/1.5086222 doi: 10.1063/1.5086222
![]() |
[14] |
B. A. Alkhaleel, H. Liao, K. M. Sullivan, Risk and resilience-based optimal post-disruption restoration for critical infrastructures under uncertainty, Eur. J. Oper. Res., 296 (2022), 174–202. https://doi.org/10.1016/j.ejor.2021.04.025 doi: 10.1016/j.ejor.2021.04.025
![]() |
[15] |
B. Cai, Y. Zhang, H. Wang, Y. Liu, R. Ji, C. Gao, et al., Resilience evaluation methodology of engineering systems with dynamic-Bayesian-network-based degradation and maintenance, Reliab. Eng. Syst. Saf., 209 (2021), 107464. https://doi.org/10.1016/j.ress.2021.107464 doi: 10.1016/j.ress.2021.107464
![]() |
[16] |
A. J. Kerkhoff, B. J. Enquist, The implications of scaling approaches for understanding resilience and reorganization in ecosystems, Bioscience, 57 (2007), 489–499. https://doi.org/10.1641/B570606 doi: 10.1641/B570606
![]() |
[17] |
Z. Chen, D. Hong, W. Cui, et al., Resilience evaluation and optimal design for weapon system of systems with dynamic reconfiguration, Reliab. Eng. Syst. Saf., 237 (2023), 109409. https://doi.org/10.1016/j.ress.2023.109409 doi: 10.1016/j.ress.2023.109409
![]() |
[18] |
Z. Chen, T. Zhao, J. Jiao, J. Chu, Performance-threshold-based resilience analysis of system of systems by considering dynamic reconfiguration, Proc. Inst. Mech. Eng., 236 (2022), 1828–1838. https://doi.org/10.1177/0954405420937528 doi: 10.1177/0954405420937528
![]() |
[19] |
S. Hosseini, D. Ivanov, A. Dolgui, Review of quantitative methods for supply chain resilience analysis, Transp. Res. Part E: Logist. Transp. Rev., 125 (2019), 285–307. https://doi.org/10.1016/j.tre.2019.03.001 doi: 10.1016/j.tre.2019.03.001
![]() |
[20] |
M. Liu, Q. Feng, D. Fan, H. Dui, B. Sun, Y. Ren, et al., Resilience importance measure and optimization considering the stepwise recovery of system performance, IEEE Trans. Reliab., 178 (2022), 178–185. https://doi.org/10.1109/TR.2022.3196058 doi: 10.1109/TR.2022.3196058
![]() |
[21] |
H. Dui, M. Liu, J. Song, S. Wu, Importance measure-based resilience management: Review, methodology and perspectives on maintenance, Reliab. Eng. Syst. Saf., 235 (2023), 109383. https://doi.org/10.1016/j.ress.2023.109383 doi: 10.1016/j.ress.2023.109383
![]() |
[22] |
S. Geng, S. Liu, Z. Fang, A demand-based framework for resilience assessment of multistate networks under disruptions, Reliab. Eng. Syst. Saf., 222 (2022) 108423. https://doi.org/10.1016/j.ress.2022.108423 doi: 10.1016/j.ress.2022.108423
![]() |
[23] |
H. Tran, M. Balchanos, J. Domerçant, D. N. Mavris, A framework for the quantitative assessment of performance-based system resilience, Reliab. Eng. Syst. Saf., 158 (2017), 73–84. https://doi.org/10.1016/j.ress.2016.10.014 doi: 10.1016/j.ress.2016.10.014
![]() |
[24] |
G. Bai, Y. Li, Y. Fang, Y. A. Zhang, J. Tao, Network approach for resilience evaluation of a UAV swarm by considering communication limits, Reliab. Eng. Syst. Saf., 193 (2020), 106602. https://doi.org/10.1016/j.ress.2019.106602 doi: 10.1016/j.ress.2019.106602
![]() |
[25] |
C. Cheng, G. Bai, Y. Zhang, J. Tao, Improved integrated metric for quantitative assessment of resilience, Adv. Mech. Eng., 12 (2020), 168–180. https://doi.org/10.1177/1687814020906065 doi: 10.1177/1687814020906065
![]() |
[26] | Q. Sun, H. Li, Y. Wang, Y. Zhang, Multi-swarm-based cooperative reconfiguration model for resilient unmanned weapon system-of-systems, Reliab. Eng. Syst. Saf., 222 (2022), 108426. https://doi.org/108426.10.1016/j.ress.2022.108426 |
[27] |
Q. Feng, M. Liu, B. Sun, H. Dui, X. Hai, Y. Ren, et al., Resilience measure and fformation reconfiguration optimization for multi-UAV systems, IEEE Internet Things J., 11 (2024), 10616–10626. https://doi.org/10.1109/JIOT.2023.3326552 doi: 10.1109/JIOT.2023.3326552
![]() |
[28] |
H. T. Tran, J. C. Domerçant, D. N. Mavris, A network-based cost comparison of resilient and robust system-of-systems, Procedia Comput. Sci., 95 (2016), 126–133. https://doi.org/10.1016/j.procs.2016.09.302 doi: 10.1016/j.procs.2016.09.302
![]() |
[29] |
X. Pan, H. Wang, Y. Yang, G. Zhang, Resilience based importance measure analysis for SoS, J. Syst. Eng. Electron., 30 (2019), 920–930. https://doi.org/10.21629/JSEE.2019.05.10 doi: 10.21629/JSEE.2019.05.10
![]() |
[30] |
Y. Cheng, E. A. Elsayed, Z. Huang, Systems resilience assessments: a review, framework and metrics, Int. J. Prod. Res., 60 (2022), 595–622. https://doi.org/10.1080/00207543.2021.1971789 doi: 10.1080/00207543.2021.1971789
![]() |
[31] |
M. Versaci, G. Angiulli, P. Crucitti, D. D. Carlo, F. Laganà, D. Pellicanò, et al., A fuzzy similarity-based approach to classify numerically simulated and experimentally detected carbon fiber-reinforced polymer plate defects, Sensors, 22 (2022), 4232. https://doi.org/10.3390/s22114232 doi: 10.3390/s22114232
![]() |
1. | Ammara Nosheen, Maria Tariq, Khuram Ali Khan, Nehad Ali Shah, Jae Dong Chung, On Caputo Fractional Derivatives and Caputo–Fabrizio Integral Operators via (s, m)-Convex Functions, 2023, 7, 2504-3110, 187, 10.3390/fractalfract7020187 | |
2. | Jie Li, Yong Lin, Serap Özcan, Muhammad Shoaib Saleem, Ahsan Fareed Shah, A study of Hermite-Hadamard inequalities via Caputo-Fabrizio fractional integral operators using strongly $(s, m)$-convex functions in the second sense, 2025, 2025, 1029-242X, 10.1186/s13660-025-03266-x |