Open AccessSome inequalities for strongly $(p,h)$-harmonic convex functions
Author(s) -
Muhammad Aslam Noor,
Khalida İnayat Noor,
Sabah Iftikhar
Publication year - 2019
Publication title -
karpatsʹkì matematičnì publìkacìï
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.63
H-Index - 4
eISSN - 2313-0210
pISSN - 2075-9827
DOI - 10.15330/cmp.11.1.119-135
Subject(s) - mathematics , convex function , harmonic , harmonic function , regular polygon , proper convex function , hermite polynomials , hypergeometric function , convex optimization , pure mathematics , convex combination , subharmonic function , hadamard transform , mathematical analysis , physics , geometry , quantum mechanics
In this paper, we show that harmonic convex functions $f$ is strongly $(p, h)$-harmonic convex functions if and only if it can be decomposed as $g(x) = f(x) - c (\frac{1}{x^p})^2,$ where $g(x)$ is $(p, h)$-harmonic convex function. We obtain some new estimates class of strongly $(p, h)$-harmonic convex functions involving hypergeometric and beta functions. As applications of our results, several important special cases are discussed. We also introduce a new class of harmonic convex functions, which is called strongly $(p, h)$-harmonic $\log$-convex functions. Some new Hermite-Hadamard type inequalities for strongly $(p, h)$-harmonic $log$-convex functions are obtained. These results can be viewed as important refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.