Some inequalities for strongly $(p,h)$-harmonic convex functions
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 co...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Vasyl Stefanyk Precarpathian National University
2019-06-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/1514 |
| Summary: | 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. |
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| ISSN: | 2075-9827 2313-0210 |