Some inequalities for the q-beta and the q-gamma functions via some q-integral inequalities

Some new inequalities for the q-gamma, the q-beta and the q-analogue of the Psi functions are established via some q-integral inequalities. In classical analysis, integral inequalities have been well-developed and leading to a wide variety of applications in mathematics and physics (see [11–13] and references therein). In a survey paper [4], Dragomir et al. used certain clever integral inequalities to provide some interesting inequalities for the Euler's beta and gamma functions. Interested by this type of inequalities, Agarwal et al. gave in [1] some improvements and generalizations of some of the Dragomir's ones. In quantum-calculus, in spite of the natural difficulties, coming from the definition of the q-Jackson integral, the interest in the q-integral inequalities has grown in the last few years (see [2,6,14]). It is within this framework that this paper presents itself. The main object is provide some new q-integral inequalities and, as applications, we establish some inequalities for the q-beta and the q-gamma functions. This paper is organized as follows: in Section 2, we present some standard conventional notations and notions which will be used in the sequel. In Section 3, we state q-analogues of the C ebys ev's integral inequalities for synchronous (asynchro-nous) mappings and as a direct consequence, we give some inequalities involving the q-beta and the q-gamma functions. In Section 4, we establish some inequalities for these functions via q-Hölder's integral inequality. Section 5 is devoted to some applications of the q-Grüss' integral inequality. Finally, Section 6 shows a q-analogue of a C ebys ev's type inequality and gives some related applications for the q-beta and the q-gamma functions. 2. Notations and preliminaries For the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper. All of these results can be found in [5,8] or [9]. Throughout this paper, we will fix q 2Š0; 1½. For a 2 C, we write ½aŠ q ¼ 1 À q a