• Title/Summary/Keyword: Hermite-Hadamard's integral inequality

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HYPERBOLIC TYPE CONVEXITY AND SOME NEW INEQUALITIES

  • Toplu, Tekin;Iscan, Imdat;Kadakal, Mahir
    • Honam Mathematical Journal
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    • v.42 no.2
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    • pp.301-318
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    • 2020
  • In this paper, we introduce and study the concept of hyperbolic type convexity functions and their some algebraic properties. We obtain Hermite-Hadamard type inequalities for this class of functions. In addition, we obtain some refinements of the Hermite-Hadamard inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is hyperbolic convexity. Moreover, we compare the results obtained with both Hölder, Hölder-İşcan inequalities and power-mean, improved-power-mean integral inequalities.

INEQUALITIES OF HERMITE-HADAMARD TYPE FOR n-TIMES DIFFERENTIABLE ARITHMETIC-HARMONICALLY FUNCTIONS

  • Kadakal, Huriye
    • Honam Mathematical Journal
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    • v.44 no.2
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    • pp.244-258
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    • 2022
  • In this work, by using an integral identity together with both the Hölder and the power-mean integral inequalities we establish several new inequalities for n-times differentiable arithmetic-harmonically-convex function. Then, using this inequalities, we obtain some new inequalities connected with means. In special cases, the results obtained coincide with the well-known results in the literature.

APPROXIMATE CONVEXITY WITH RESPECT TO INTEGRAL ARITHMETIC MEAN

  • Zoldak, Marek
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1829-1839
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    • 2014
  • Let (${\Omega}$, $\mathcal{S}$, ${\mu}$) be a probabilistic measure space, ${\varepsilon}{\in}\mathbb{R}$, ${\delta}{\geq}0$, p > 0 be given numbers and let $P{\subset}\mathbb{R}$ be an open interval. We consider a class of functions $f:P{\rightarrow}\mathbb{R}$, satisfying the inequality $$f(EX){\leq}E(f{\circ}X)+{\varepsilon}E({\mid}X-EX{\mid}^p)+{\delta}$$ for each $\mathcal{S}$-measurable simple function $X:{\Omega}{\rightarrow}P$. We show that if additionally the set of values of ${\mu}$ is equal to [0, 1] then $f:P{\rightarrow}\mathbb{R}$ satisfies the above condition if and only if $$f(tx+(1-t)y){\leq}tf(x)+(1-t)f(y)+{\varepsilon}[(1-t)^pt+t^p(1-t)]{\mid}x-y{\mid}^p+{\delta}$$ for $x,y{\in}P$, $t{\in}[0,1]$. We also prove some basic properties of such functions, e.g. the existence of subdifferentials, Hermite-Hadamard inequality.

ON IMPROVEMENTS OF SOME INTEGRAL INEQUALITIES

  • Kadakal, Mahir;Iscan, Imdat;Kadakal, Huriye;Bekar, Kerim
    • Honam Mathematical Journal
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    • v.43 no.3
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    • pp.441-452
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    • 2021
  • In this paper, improved power-mean integral inequality, which provides a better approach than power-mean integral inequality, is proved. Using Hölder-İşcan integral inequality and improved power-mean integral inequality, some inequalities of Hadamard's type for functions whose derivatives in absolute value at certain power are quasi-convex are given. In addition, the results obtained are compared with the previous ones. Then, it is shown that the results obtained together with identity are better than those previously obtained.

HERMITE-HADAMARD TYPE INEQUALITIES FOR GEOMETRIC-ARITHMETICALLY s-CONVEX FUNCTIONS

  • Hua, Ju;Xi, Bo-Yan;Qi, Feng
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.51-63
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    • 2014
  • In the paper, several properties of geometric-arithmetically s-convex functions are provided, an integral identity in which the integrands are products of a function and a derivative is found, and then some inequalities of Hermite-Hadamard type for integrals whose integrands are products of a derivative and a function whose derivative is of the geometric-arithmetic s-convexity are established.

EXTENDED HERMITE-HADAMARD(H-H) AND FEJER'S INEQUALITIES BASED ON GEOMETRICALLY-s-CONVEX FUNCTIONS IN THIRD AND FOURTH SENSE

  • SABIR YASIN;MASNITA MISIRAN;ZURNI OMAR;RABIA LUQMAN
    • Journal of applied mathematics & informatics
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    • v.41 no.5
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    • pp.963-972
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    • 2023
  • In this paper, geometrically convex and s-convex functions in third and fourth sense are merged to form (g, s)-convex function. Characterizations of (g, s)-convex function, algebraic and functional properties are presented. In addition, novel functions based on the integral of (g, s)-convex functions in the third sense are created, and inequality relations for these functions are explored and examined under particular conditions. Further, there are also some relationships between (g, s)-convex function and previously defined functions. The (g, s)-convex function and its derivatives will then be used to extend the well-known H-H and Fejer's type inequalities. In order to obtain the previously mentioned conclusions, several special cases from previous literature for extended H-H and Fejer's inequalities are also investigated. The relation between the average (mean) values and newly created H-H and Fejer's inequalities are also examined.

TIME SCALES INTEGRAL INEQUALITIES FOR SUPERQUADRATIC FUNCTIONS

  • Baric, Josipa;Bibi, Rabia;Bohner, Martin;Pecaric, Josip
    • Journal of the Korean Mathematical Society
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    • v.50 no.3
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    • pp.465-477
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    • 2013
  • In this paper, two different methods of proving Jensen's inequality on time scales for superquadratic functions are demonstrated. Some refinements of classical inequalities on time scales are obtained using properties of superquadratic functions and some known results for isotonic linear functionals.

CERTAIN GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR LOCAL FRACTIONAL INTEGRALS

  • Choi, Junesang;Set, Erhan;Tomar, Muharrem
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.601-617
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    • 2017
  • We give a function associated with generalized Ostrowski type inequality and its integral representation for local fractional calculus. Then, using this function and its integral representation, we establish several inequalities of generalized Ostrowski type for twice local fractional differentiable functions. We also consider some special cases of the main results which are further applied to a concrete function to yield two interesting inequalities associated with two generalized means.