Generalized harmonically convex functions on fractal sets and related Hermite-Hadamard type inequalities

Self Cite

In this paper, we establish a local fractional integral identity with a parameter λ on Yang's fractal sets. Using this identity, by generalized power mean inequality and generalized Hölder inequality, two Hermite‐Hadamard type local fractional integral inequalities for generalized harmonically convex functions are established. By giving some special values to the parameter, some inequalities with specific form can be obtained. Some applications to generalized special means are given.

s

‐convex function on Yang's fractal sets, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Mo et al in introduced generalized convex function and generalized

s

‐convex function on Yang's fractal sets, respectively. Sun introduced generalized harmonically convex function on Yang's fractal set

ℜ^{ξ} false(0 < ξ ⩽ 1 false)

, and proved Hermite‐Hadamard's inequality for generalized harmonically convex function.…”

Section: Introductionmentioning

confidence: 99%

“…Definition Let

normalΥ \subset false(0, \infty false)

be an interval. for all

u, v \in normalΥ

τ \in false[0, 1 false]

we have

g ((), \frac{u v}{τ u + false(1 - τ false) v}) ⩽ τ^{ξ} g false(v false) + {false(1 - τ false)}^{ξ} g false(u false),

then

g : normalΥ \to ℜ^{ξ} false(0 < ξ ⩽ 1 false)

is called a generalized harmonically convex function.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem Let

g : normalΥ \subset false(0, \infty false) \to ℜ^{ξ}

be a generalized harmonically convex function on Yang's fractal sets,

u, v \in normalΥ

and

u < v

. If

g false(x false) \in {scriptI}_{x}^{false(ξ false)} false[u, v false]

, then

\frac{1}{normalΓ false(1 + ξ false)} g ((), \frac{2 u v}{u + v}) ⩽ \frac{u^{ξ} v^{ξ}}{{false(v - u false)}^{ξ}}_{u} {scriptI}_{v}^{false(ξ false)} \frac{g false(x false)}{x^{2 ξ}} ⩽ \frac{normalΓ false(1 + ξ false)}{normalΓ false(1 + 2 ξ false)} ([], g false(u false) + g false(v false)) .

…”

Section: Introductionmentioning

confidence: 99%

“…Proposition If

g : false(0, \infty false) \to ℜ^{ξ}

is generalized convex and nondecreasing on

false(0, \infty false)

, then

g

is generalized harmonically convex.…”

Section: Introductionmentioning

confidence: 99%

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Hadamard type local fractional integral inequalities for generalized harmonically convex functions and applications

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Liu

2020

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Some new inequalities for generalized h‐convex functions involving local fractional integral operators with Mittag‐Leffler kernel

Sun

2020

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In this paper, we firstly construct two local fractional integral operators with Mittag‐Leffler kernel on Yang's fractal sets. Then, two local fractional integral identities with the first‐ and second‐order derivatives are derived. With these as auxiliary tools, we establish some new Hermite‐Hadamard–type local fractional integral inequalities involving the local fractional integral operators with Mittag‐Leffler kernel for generalized h‐convex functions. In addition, we obtain some special inequalities when the parameter β and function h take special values. Finally, two examples are given to illustrate the application of the results.

A novel comprehensive analysis on generalized harmonically ψ‐convex with respect to Raina's function on fractal set with applications

Chu

Rashid

Singh

2021

The pivotal proposal of this work is to present a new class of harmonically convex functions, namely, generalized harmonically 𝜓-convex functions based on the fractal set technique for establishing inequalities of Hermite-Hadamard type, Pachpatte type, and certain related variants with respect to the Raina's function. With the aid of an auxiliary identity correlated with Raina's function, by generalized Hölder inequality, two Hermite-Hadamard-type local fractional integral inequalities for generalized harmonically 𝜓-convex functions are apprehended.The proposed technique provides the results by giving some special values to the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.