In this paper an extended generalized Mittag-Leffler function $\begin{array}{} \displaystyle E_{\rho,\sigma,\tau}^{\delta,r,q,c}(z;p) \end{array}$ and the corresponding fractional integral operator $\begin{array}{} \displaystyle \varepsilon_{a^{+},\rho,\sigma,\tau}^{w,\delta,q,r,c}f \end{array}$ are defined and used to obtain generalizations of Opial-type inequalities due to Mitrinović and Pečarić. Also, some interesting properties of this function and its integral operator are discussed. Several known results are deduced.

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Integral operators are useful in real analysis, mathematical analysis, functional analysis and other subjects of mathematical approach. The goal of this paper is to study a unified integral operator via convexity. By using convexity and conditions of unified integral operators, bounds of these operators are obtained. Furthermore consequences of these results are discussed for fractional and conformable integral operators.

In this article, we establish bounds of sum of the left and right sided Riemann Liouville (RL) fractional integrals and related inequalities in general form. A new and novel approach is followed to obtain these results for general Riemann Liouville (RL) fractional integrals. Monotonicity and convexity of functions are used with some usual and straight forward inequalities. The presented results are also have connection with some known and already published results. Applications and motivations of presented results are briefly discussed.

Abstract. In this paper we prove the Hadamard-type inequalities for m-convex functions via Riemann-Liouville fractional integrals and the Hadamard-type inequalities for convex functions via Riemann-Liouville fractional integral are deduced. Also we find connections with some well known results related to the Hadamard inequality.Mathematics Subject Classification (2010): 26A51, 26A33, 26D10.