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New fractal-fractional parametric inequalities with applications

CHAOS SOLITONS & FRACTALS 2023年第1期172卷

作者： Butt, Saad Ihsan Khan, Ahmad COMSATS Univ Islamabad Dept Math Lahore Campus Islamabad Pakistan

In the present paper, we first establish a general parameterized identity for local fractional twice differentiable functions involving extended fractal-fractional integral operators. Thus by employing generalized convexity on differentiable mappings along with Yang's Power-mean, Holder's and improved fractal integral inequalities lead us to develop variety of new fractal-fractional parameterized inequalities. Several examples are provided with graphical illustrations to prove the validity of new results. We give error analysis of improved bounds numerically and also by 2D, 3D graphical representations. Finally, we show that our main results recapture fractal variants of trapezoid, midpoint, Simpson and Bullen-type inequalities. Some related applications to the fractal means, moment of random variables and wave equations are given as well.

关键词： Fractal theory generalized convex functions Quadrature inequalities generalized fractional integral operators

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NEWTON'S-TYPE INTEGRAL INEQUALITIES VIA LOCAL FRACTIONAL INTEGRALS

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FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2020年第3期28卷

作者： Iftikhar, Sabah Kumam, Poom Erden, Samet King Mongkuts Univ Technol Thonburi KMUTT Ctr Excellence Theoret & Computat Sci TaCS CoE SCL Fixed Point Lab 802 Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung 40402 Taiwan Bartin Univ Fac Sci Dept Math Bartin Turkey

We firstly establish an identity involving local fractional integrals. Then, with the help of this equality, some new Newton-type inequalities for functions whose the local fractional derivatives in modulus and their some powers are generalized convex are obtained. Some applications of these inequalities for Simpson's quadrature rules and generalized special means are also given.

关键词： Newton's Inequality generalized convex functions Local Fractional Integrals

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generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property

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ADVANCES IN DIFFERENCE EQUATIONS 2020年第1期2020卷 1-24页

作者： Khan, Zareen A. Rashid, Saima Ashraf, Rehana Baleanu, Dumitru Chu, Yu-Ming Princess Nourah Bint Abdulrahman Univ Coll Sci Dept Math Riyadh Saudi Arabia Govt Coll Univ Dept Math Faisalabad Pakistan Lahore Coll Women Univ Dept Math Lahore Pakistan Cankaya Univ Fac Arts & Sci Dept Math Ankara 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

In the paper, we extend some previous results dealing with the Hermite-Hadamard inequalities with fractal sets and several auxiliary results that vary with local fractional derivatives introduced in the recent literature. We provide new generalizations for the third-order differentiability by employing the local fractional technique for functions whose local fractional derivatives in the absolute values are generalized convex and obtain several bounds and new results applicable to convex functions by using the generalized Holder and power-mean *** an application, numerous novel cases can be obtained from our outcomes. To ensure the feasibility of the proposed method, we present two examples to verify the method. It should be pointed out that the investigation of our findings in fractal analysis and inequality theory is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.

关键词： generalized convex functions Hermite-Hadamard inequalty ebyev inequality generalized Holder inequality Power-mean inequality Fractal set 26D15 26D10 90C23

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On certain properties of hyperbolically convex functions

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS

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ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2019年第42期 476-484页

作者： Faried, Nashat Ali, Mohamed S. S. Yehia, Zeinab M. Ain Shams Univ Fac Sci Dept Math Cairo Egypt Ain Shams Univ Fac Educ Dept Math Cairo Egypt

The aim of this paper is to prove that the envelope of hyperbolically convex functions is hyperbolically convex function. Furthermore, we study the standard functional operations of hyperbolically convex functions and... 详细信息

关键词： generalized convex functions hyperbolically convex functions supporting functions majorization

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New Inequalities for Local Fractional Integrals

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IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY TRANSACTION A-SCIENCE 2017年第A4期41卷 1039-1046页

作者： Budak, Huseyin Sarikaya, Mehmet Zeki Yildirim, Huseyin Duzce Univ Fac Sci & Arts Dept Math Duzce Turkey Kahramanmaras Sutcu Imam Univ Fac Sci & Arts Dept Math Kahramanmaras Turkey

In this paper, we establish double inequalities for twice local fractional differentiable mappings on fractal sets R-alpha(0 < alpha <= 1) of real line numbers. We also give some generalized Hermite-Hadamard lik... 详细信息

关键词： Local fractional integral Fractal space generalized Hermite-Hadamard inequality generalized convex functions generalized Gruss inequality

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Fractal integral inequalities for harmonic convex functions

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Applied Mathematics and Information Sciences 2018年第4期12卷 831-839页

作者： Noor, Muhammad Aslam Noor, Khalida Inayat Iftikhar, Sabah Awan, Muhammad Uzair Mathematics Department COMSATS University Islamabad Islamabad Pakistan Department of Mathematics GC University Faisalabad Pakistan

In this paper, we introduce a new class of convex functions, which is called generalized harmonic convex functions on fractal sets ℝα (0 α ≤ 1). This new class of convex functions includes generalized convex functi... 详细信息

In this paper, we introduce a new class of convex functions, which is called generalized harmonic convex functions on fractal sets ℝ^α (0 α ≤ 1). This new class of convex functions includes generalized convex functions and harmonic convex functions as special *** obtain several new Hermite-Hadamard inequalities for generalized harmonic convex functions. Some special cases are also discussed. The ideas and technique of this paper may motivate further research in this field. © 2018 NSP.

关键词： generalized convex functions generalized Hermite-Hadamard inequality Harmonic convex functions

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Higher order optimality conditions with an arbitrary non-differentiable function

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OPTIMIZATION 2016年第11期65卷 1909-1927页

作者： Ivanov, Vsevolod I. Tech Univ Varna Dept Math Varna Bulgaria

In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real function. We obtain necessary and sufficient optimality conditions of order n (n is a positive integer) for unconstrained problems in terms of them. We do not require any restrictions on the function in our results. In contrast to the most known directional derivatives, our derivative is harmonized with the classical higher order Frechet directional derivative of the same order in the sense that both of them coincide, provided that the last one exists. A notion of a higher order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order n. Higher order invex functions are defined. They are the largest class such that the necessary conditions for a local minimum are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order n.

关键词： Non-smooth optimization higher order directional derivatives of Hadamard type higher order subdifferentials necessary and sufficient conditions for optimality generalized convex functions 90C46 49K10 26B05 26B25

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Second-order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2015年 125卷 270-289页

作者： Ivanov, Vsevolod I. Tech Univ Varna Dept Math Varna 9010 Bulgaria

In this paper, we introduce a new second-order directional derivative and a second-order subdifferential of Hadamard type for an arbitrary nondifferentiable function. We derive second-order necessary and sufficient optimality conditions for a local and a global minimum and an isolated local minimum of second-order in unconstrained optimization. In particular, we obtain two results with strongly pseudoconvex functions. We also compare our conditions with the results of the recently published paper (Bednarik and Pastor, 2008) and a lot of other works, published in high level journals, and prove that they are particular cases of our necessary and sufficient ones. We prove that the necessary optimality conditions concern more functions than the conditions in terms of lower Dini directional derivative, even the optimality conditions with the last derivative can be applied to a function, which does not belong to some special class. At last, we apply our optimality criteria for scalar problems to derive necessary and sufficient optimality conditions in the cone-constrained vector optimization. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Nonsmooth optimization Second-order isolated minimizers Strict local minimizers of order 2 Optimality conditions generalized convex functions

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generalized MONOTONE OPERATORS ON DENSE SETS

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2015年第7期36卷 901-929页

作者： Laszlo, Szilard Viorel, Adrian Tech Univ Cluj Napoca Dept Math Cluj Napoca 400114 Romania

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.

关键词： generalized convex functions generalized monotone operators Locally generalized monotone functions Self segment-dense sets 47H04 47H05 26B25 26E25

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Optimality conditions for multiple objective fractional subset programming with (ρ,σ,θ)-type-I and related non-convex functions

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CALCOLO 2012年第3期49卷 177-192页

作者： Mishra, S. K. Jaiswal, M. Pankaj Banaras Hindu Univ Dept Math Fac Sci DST Ctr Interdisciplinary Math Sci Varanasi 221005 Uttar Pradesh India Banaras Hindu Univ Mahila Maha Vidhyalaya Varanasi 221005 Uttar Pradesh India

In this paper, we introduce a class of generalized invex n-set functions, called (rho,sigma,theta)-type-I and related non-convex functions, and then establish a number of parametric and semi-parametric sufficient optimality conditions for the primal problem under the aforesaid assumptions. This work partially extends an earlier work of Mishra et al. (Math. Methods Oper. Res. 67, 493-504, 2008) to a wider class of functions.

关键词： Multiple objective fractional subset programming Sufficient optimality conditions generalized convex functions

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