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New Multi-Parametrized Estimates Having pth-Order Differentiability in Fractional Calculus for Predominating -Convex Functions in Hilbert Space

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Department of Mathematics, Government College University, Faisalabad 38000, Pakistan
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School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
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Faculty of Science and Techniques Moulay Ismail University of Meknes, 52000 Errachidia, Morocco
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Department of Mathematics, Lahore College for Women University, Jhangh Campus, Lahore 54500, Pakistan
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Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey
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Department of Mathematics, Huzhou University, Huzhou 313000, China
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Author to whom correspondence should be addressed.
Symmetry 2020, 12(2), 222; https://doi.org/10.3390/sym12020222
Received: 26 December 2019 / Revised: 15 January 2020 / Accepted: 19 January 2020 / Published: 2 February 2020
In Hilbert space, we develop a novel framework to study for two new classes of convex function depending on arbitrary non-negative function, which is called a predominating -convex function and predominating quasiconvex function, with respect to η , are presented. To ensure the symmetry of data segmentation and with the discussion of special cases, it is shown that these classes capture other classes of η -convex functions, η -quasiconvex functions, strongly -convex functions of higher-order and strongly quasiconvex functions of a higher order, etc. Meanwhile, an auxiliary result is proved in the sense of κ -fractional integral operator to generate novel variants related to the Hermite–Hadamard type for p t h -order differentiability. It is hoped that this research study will open new doors for in-depth investigation in convexity theory frameworks of a varying nature.
Keywords: convex function; η-convex functions; predominating convex functions; Hermite–Hadamard inequality; predominating η-quasiconvex functions convex function; η-convex functions; predominating convex functions; Hermite–Hadamard inequality; predominating η-quasiconvex functions
MDPI and ACS Style

Rashid, S.; Kalsoom, H.; Hammouch, Z.; Ashraf, R.; Baleanu, D.; Chu, Y.-M. New Multi-Parametrized Estimates Having pth-Order Differentiability in Fractional Calculus for Predominating -Convex Functions in Hilbert Space. Symmetry 2020, 12, 222.

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