### ON PARAMETRIZED HERMITE-HADAMARD TYPE INEQUALITIES

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#### Abstract

In recent years, many results have been devoted to the well-known Hermite-Hadamard inequality. This inequality has many applications in the area of pure and applied mathematics. In this paper, our main aim is to give a parametrized inequality of the Hermite-Hadamard type and its applications to f-divergence measures and means.First, we prove the identity associated with the right side of the Hermite-Hadamard inequality. By using this identity, the convexity of the function and some well-known inequalities, we obtain several results for the inequality. The inequalities derived here also point out some known results as their special cases.

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S. M. Ali and S. D. Silvey: A general class of coefficients of divergence of one

distribution from another. J. Roy. Statist. Soc. Sec B 28 (1996), 131-142.

R. F. Bai, F. Qi and B. Y. Xi: Hermite-Hadamard type inequalities for the m-

and (α,m)-logarithmically convex functions. Filomat 27(1) (2013), 1-7.

A. Barani, A. G. Ghazanfari, and S. S. Dragomir: Hermite-Hadamard in-

equality for functions whose derivatives absolute values are preinvex. J. Inequal.

Appl. 2012(6) (2012), 247.

S. Belarbi and Z. Dahmani: On some new fractional integral inequalities. J.

Inequal. Pure Appl. Math. 10(3) (2009), 85-86.

I. Burbea and C. R, Rao: On the convexity of some divergence measures based

on entropy function. IEEE Trans. Inf. Th. 28(3) (1982), 489-495.

Y. M. Chu, M. A. khan, T. Ali and S. S. Dragomir: Inequalities for α-

fractional differentiable functions. J. Inequal. Appl. 93 (2017), 12 pages.

Y. M. Chu, M. A. khan, M. TU. Khan and T. Ali: Generalizations of Hermite-

Hadamard type inequalities for MT-convex functions. J. Nonlinear Sci. Appl. 9(6)

(2016), 4305-4316.

I. Csis´ zar: Information-type measures of difference of probability distributions

and indirect observations. Studia Math. Hungarica 2 (1967), 299-318.

S. S. Dragomir : Two mappings in connection to Hadamard’s inequality. J.

Math. Anal. Appl. 167 (1992), 49-56.

S. S. Dragomir and R. P. Agarwal: Two inequalities for differentiable map-

pings and applications to special means of real numbers and to Trapezoidal for-

mula. Appl. Math. Lett. 11(5) (1998), 91-95.

S. S. Dragomir and S. Fitzpatrick: The Hadamard inequalities for s-convex

functions in the second sense. Demonstratio Math. 32(4) (1999), 687-696.

S. S. Dragomir and A. Mcandrew: Refinements of the Hermite-Hadamard

inequality for convex functions. J. Inequal. Pure Appl. Math. 6 (2005), 1-6.

S. S. Dragomir and C. E. M. Pearce: Selected topics on Hermite-Hadamard

inequalities and applications. Victoria University, 2000.

S. S. Dragomir, J. Pecaric and L. E. Persson: Some inequalities of Hadamard

type. Soochow J. Math. 21 (1995), 335-341.

J. Hadamard: ’Etude sur les propri´ et´ es des fonctions enti` eres et en particulier

`

dune fonction consid´ er´ ee par Riemann. J. Math. Pures Appl. 58 (1893), 171-215.

J. H. Havrda and F. Charvat: Quantification method classification process.

concept of structural-entropy, Kybernetika 3 (1967), 30-35.

J. N. Kapur: A comparative assessment of various measures of directed diver-

gence. Advances in Management Studies 3 (1984), 1-16.

H. Kavurmaci, M. Avci and M. E. Ozdemir: New inequalities of hermite-

hadamard type for convex functions with applications. J. Inequal. Appl. 86 (2010),

-11

M. A. khan, T. Ali, S. S. Dragomir and M. Z. Sarikaya: HermiteHadamard

type inequalities for conformable fractional integrals. Rev. R. Acad. Cienc. Exactas

Fs. Nat. Ser. AMat. (2017).

M. A. khan, Y. Khurshid and T. Ali: Hermite-Hadamard inequality for frac-

tional integrals via η-convex functions. Acta Math. Univ. Comenian. 86(1) (2017),

-164.

M. A. khan, Y. Khurshid and T. Ali: Inequalities for three times differential

functions. Acta Math. Univ. Comenian. 48(2) (2016), 1-14.

M. A. khan, Y. Khurshid, T.S. Du and Y.M. Chu: Generalization of Hermite-

Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces

(2018), 1-12.

M. A. khan, Y. Khurshid, S. S. Dragomir and R. Ullah: New Hermite-

Hadamard type inequalities with applications. Punjab Univ. J. Math. 50(3) (2018),

-12.

U. S. Kirmaci, M. K. Bakula, M. E. Ozdemir and J. Pecaric: Hadamard-

type inequalities for s-convex functions. Appl. Math. and Comput. 193 (2007),

-35.

S. Kullback and R. A. Leibler: On information and sufficiency. Ann. Math.

Stat. 22 (1951), 79-86.

J. Lin: Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Th.

(1) (1991), 145-151.

M. E. Ozdemir, M. Avci and E. Set: On some inequalities of HermiteHadamard

type via m-convexity. App. Math. Lett. 23(9) (2010), 1065-1070.

C. E. M. Pearce and J. Peˇ cari´ c: Inequalities for diffrentiable mapping with

application to special means and quadrature formula. Appl. Math. Lett. 13 (2000),

-55.

A. Renyi: On measures of entropy and information. Proc. Fourth Berkeley Symp.

Math. Stat. and Prob., University of California Press 1 (1961), 547-561.

M. Z. Sarikaya, A. Saglam and H. Yildrim: On some Hadamard-type inequal-

ities for h-convex functions. J. Math. Inequal. 2 (2008), 335-341.

H. Shioya and T. Da-Te: A generalisation of Lin divergence and the deriva

DOI: https://doi.org/10.22190/FUMI1902213K

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