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Subharmonic functions with Separated Variables and Their Connection with generalized convex functions

RUSSIAN MATHEMATICS 2024年第6期68卷 42-57页

作者： Muryasov, R. R. Russian Acad Sci Inst Math Comp Ctr Subdiv Ufa Fed Res Ctr Ufa 450008 Russia

In this paper, we consider the necessary and sufficient conditions for the subharmonicity of functions of two variables representable as a product of two functions of one variable in the Cartesian coordinate system or in the polar coordinate system in domains on the plane. We establish a connection of such functions with functions that are convex with respect to solutions of second-order linear differential equations, that is, convex with respect to two functions.

关键词： subharmonic function generalized derivative distribution convex function differential operator generalized convex function

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Genuine Bernstein-Durrmeyer type operators preserving 1 and x^j

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ANNALS OF functionAL ANALYSIS 2024年第1期15卷 1-22页

作者： Abel, Ulrich Acu, Ana Maria Heilmann, Margareta Rasa, Ioan Tech Hsch Mittelhessen Fachbereich MND Giessen Germany Lucian Blaga Univ Sibiu Dept Math & Informat Sibiu Romania Univ Wuppertal Sch Math & Nat Sci Wuppertal Germany Tech Univ Cluj Napoca Dept Math Cluj Napoca Romania

We introduce a family of genuine Bernstein-Durrmeyer type operators preserving the functions 1 and x(j). For them, we establish Voronovskaja type formulas. The behaviour with respect to generalized convex functions is... 详细信息

关键词： Genuine Bernstein-Durrmeyer operator Aldaz-Kounchev-Render operator Voronovskaya type formulas generalized convex function

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Optimality Conditions for generalized convex Nonsmooth Uncertain Multi-objective Fractional Programming

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Journal of the Operations Research Society of China 2023年第4期11卷 809-826页

作者： Xiao Pan Guo-Lin Yu Tian-Tian Gong School of Mathematics and Information Science North Minzu UniversityYinchuan750021NingxiaChina

This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex function pairs,called type-I functions and pseudo-quasi-type-I functions,are introduced in this paper for(NUMFP).Under the assumption that(NUMFP)satisfies the robust constraint qualification with respect to Clarke subdifferential,necessary optimality conditions of the robust weak efficient solution are *** optimality conditions are obtained under pseudo-quasi-type-I generalized convexity ***,we introduce the concept of robust weak saddle points to(NUMFP),and prove two theorems about robust weak saddle *** main results in the present paper are verified by concrete examples.

关键词： Multi-objective fractional programming Robust weak efficient solution generalized convex function Optimality condition Saddle point

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An extension of the proximal point algorithm beyond convexity

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JOURNAL OF GLOBAL OPTIMIZATION 2022年第2期82卷 313-329页

作者： Grad, Sorin-Mihai Lara, Felipe Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Univ Tarapaca Fac Ciencias Dept Matemat Arica Chile

We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.

关键词： Nonsmooth optimization Nonconvex optimization Proximity operator Proximal point algorithm generalized convex function

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generalized integral inequalities for convex functions on the co-ordinates

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TBILISI MATHEMATICAL JOURNAL 2021年第3期14卷 81-94页

作者： Butt, Saad Ihsan Kashuri, Artion Nadeem, Muhammad COMSATS Univ Islamabad Lahore Campus Lahore Pakistan Univ Ismail Qemali Fac Tech Sci Dept Math Vlora 9400 Albania

The aim of this paper is to establish some new inequalities of Hermite-Hadamard and Fejer type using generalized h-convex and generalized (h(1), h(2))-convex functions on the co-ordinates. Some known results are recap... 详细信息

关键词： Hermite-Hadamard inequality Hermite-Hadamard-Fejer inequality coordinated convex function generalized convex function Raina's function

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JENSEN-MERCER INEQUALITY AND RELATED RESULTS IN THE FRACTAL SENSE WITH APPLICATIONS

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FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2022年第1期30卷 2240008-2240008页

作者： Butt, Saad Ihsan Yousaf, Saba Ahmad, Hijaz Nofal, Taher A. COMSATS Univ Islamabad Dept Math Lahore Campus Pakistan Lahore Pakistan Int Telemat Univ Uninettuno Sect Math Corso Vittorio Emanuele II39 I-00186 Rome Italy Al Ayen Univ Sci Res Ctr Math Appl Sci & Engn Res Grp Thi Qar 64001 Iraq Taif Univ Dept Math Coll Sci POB 11099 At Taif 21944 Saudi Arabia

The most notable inequality pertaining convex functions is Jensen's inequality which has tremendous applications in several fields. Mercer introduced an important variant of Jensen's inequality called as Jensen-Mercer's inequality. Fractal sets are useful tools for describing the accuracy of inequalities in convex functions. The purpose of this paper is to establish a generalized Jensen-Mercer inequality for a generalized convex function on a real linear fractal set Double-struck capital R alpha (0 < alpha <= 1). Further, we also demonstrate some generalized Jensen-Mercer-type inequalities by employing local fractional calculus. Lastly, some applications related to Jensen-Mercer inequality and alpha-type special means are given. The present approach is efficient, reliable, and may motivate further research in this area.

关键词： Jensen-Mercer Inequality Fractal Space generalized convex function generalized Jensen Inequality Local Fractional Derivative Local Fractional Integral

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FRACTAL HADAMARD-MERCER-TYPE INEQUALITIES WITH APPLICATIONS

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FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2022年第2期30卷 2240055-2240055页

作者： Butt, Saad Ihsan Yousaf, Saba Younas, Muhammad Ahmad, Hijaz Yao, Shao-Wen COMSATS Univ Islamabad Dept Math Lahore Campus Lahore Pakistan Istanbul Ticaret Univ Informat Technol Applicat & Res Ctr TR-34445 Istanbul Turkey Istanbul Ticaret Univ Fac Humanities & Social Sci Dept Math TR-34445 Istanbul Turkey Henan Polytech Univ Sch Math & Informat Sci Jiaozuo 454000 Henan Peoples R China

Fractal analysis is a totally new area of research based on local fractional calculus. It has interesting applications in various fields such as a complex graph, computer graphics, the music industry, picture compression and many more fields. In this paper, we present new variants of Hadamard-Mercer-type inequalities on fractal sets R-alpha (0 < alpha <= 1) by employing generalized convex function. We establish two new lemmas involving local fractional integrals. By using these lemmas, we obtain several results related to generalized Hadamard Mercer-type integral inequalities for local differentiable generalized convex functions on real linear fractal space. Finally, we give applications for probability density functions and compute new generalized means.

关键词： Jensen-Mercer Inequality Fractal Space generalized convex function generalized Hermite-Hadarnard Inequality Local Fractional Derivative Local Fractional Integral

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Some new local fractional inequalities associated with generalized (s,m)-convex functions and applications

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ADVANCES IN DIFFERENCE EQUATIONS 2020年第1期2020卷 1-27页

作者： Abdeljawad, Thabet Rashid, Saima Hammouch, Zakia Chu, Yu-Ming Prince Sultan Univ Dept Math & Gen Sci Riyadh Saudi Arabia China Med Univ Taiwan Dept Med Res Taichung Taiwan Asia Univ Dept Comp Sci & Informat Engn Taichung Taiwan Govt Coll Univ Dept Math Faisalabad Pakistan Moulay Ismail Univ Meknes Fac Sci & Tech Errachidia Morocco Huzhou Univ Dept Math Huzhou Peoples R China Changsha Univ Sci & Technol Hunan Prov Key Lab Math Modeling & Anal Engn Changsha Peoples R China

Fractal analysis is one of interesting research areas of computer science and engineering, which depicts a precise description of phenomena in modeling. Visual beauty and self-similarity has made it an attractive field of research. The fractal sets are the effective tools to describe the accuracy of the inequalities for convex functions. In this paper, we employ linear fractals R alpha to investigate the (s,m)-convexity and relate them to derive generalized Hermite-Hadamard (HH) type inequalities and several other associated variants depending on an auxiliary result. Under this novel approach, we aim at establishing an analog with the help of local fractional integration. Meanwhile, we establish generalized Simpson-type inequalities for (s,m)-convex functions. The results in the frame of local fractional showed that among all comparisons, we can only see the correlation between novel strategies and the earlier consequences in generalized s-convex, generalized m-convex, and generalized convex functions. We obtain application in probability density functions and generalized special means to confirm the relevance and computational effectiveness of the considered method. Similar results in this dynamic field can also be widely applied to other types of fractals and explored similarly to what has been done in this paper.

关键词： generalized convex function generalized s-convex function Hermite-Hadamard inequality Simpson-type inequality generalized m-convex functions Fractal sets 26E60

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Some new Simpson-type inequalities for generalizedp-convex function on fractal sets with applications

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ADVANCES IN DIFFERENCE EQUATIONS 2020年第1期2020卷 1-26页

作者： Abdeljawad, Thabet Rashid, Saima Hammouch, Zakia Iscan, Imdat Chu, Yu-Ming Prince Sultan Univ Dept Math & Gen Sci Riyadh Saudi Arabia China Med Univ Dept Med Res Taichung Taiwan Asia Univ Dept Comp Sci & Informat Engn Taichung Taiwan Govt Coll Univ Dept Math Faisalabad Pakistan Univ Moulay Ismail Fac Sci & Tech Dept Math Errachidia Morocco Giresun Univ Fac Arts & Sci Dept Math Giresun Turkey Huzhou Univ Dept Math Huzhou Peoples R China Changsha Univ Sci Technol Hunan Prov Key Lab Math Modeling & Anal Engn Changsha Peoples R China

The present article addresses the concept ofp-convex functions on fractal sets. We are able to prove a novel auxiliary result. In the application aspect, the fidelity of the local fractional is used to establish the generalization of Simpson-type inequalities for the class of functions whose local fractional derivatives in absolute values at certain powers arep-convex. The method we present is an alternative in showing the classical variants associated with generalizedp-convex functions. Some parts of our results cover the classical convex functions and classical harmonically convex functions. Some novel applications in random variables, cumulative distribution functions and generalized bivariate means are obtained to ensure the correctness of the present results. The present approach is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractals in computer graphics.

关键词： generalized convex function generalizeds-convex function Hermite-Hadamard inequality Simpson's-like type inequality generalizedm-convex functions Fractal sets

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NEW NEWTON'S TYPE ESTIMATES PERTAINING TO LOCAL FRACTIONAL INTEGRAL VIA generalized p-convexITY WITH APPLICATIONS

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FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2021年第5期29卷 2140018-2140018页

作者： LI, Yong-min Rashid, Saima Hammouch, Zakia Baleanu, Dumitru Chu, Yu-ming Huzhou Univ Dept Math Huzhou 313000 Peoples R China Govt Coll Univ Dept Math Faisalabad 38000 Pakistan Thu Dau Mot Univ Div Appl Math Binh Duong Province Vietnam China Med Univ Dept Med Res China Med Univ Hosp Taichung 40402 Taiwan Moulay Ismail Univ Ecole Normale Super Meknes 5000 Morocco Cankaya Univ Dept Math Ankara Turkey Hunan City Univ Coll Sci Yiyang 413000 Peoples R China

This paper aims to investigate the notion of p-convex functions on fractal sets Double-struck capital R-alpha(0 < alpha <= 1). Based on these novel ideas, we derived an auxiliary result depend on a three-step quadratic kernel by employing generalized p-convexity. Take into account the local fractal identity, we established novel Newton's type variants for the local differentiable functions. Several special cases are apprehended in the light of generalized convex functions and generalized harmonically convex functions. This novel strategy captures several existing results in the relative literature. Application is obtained in cumulative distribution function and generalized special weighted means to confirm the relevance and computational effectiveness of the considered method. Finally, we supposed that the consequences of this paper can stimulate those who are interested in fractal analysis.

关键词： generalized convex function generalized Harmonically convex function generalized p-convex functions Newton's Type Inequality Fractal Sets

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