We provide an overview of the Riemann ζ-function with an eye to a proof of the functional equation that is non-standard and more accessible than those proofs readily available in modern texts. This is s survey note with the goal of a more elegant and transparent presentation of the salient features surrounding the functional equation for the ζ-function and the intimate connections with the Riem...

Abstract In this paper, we establish some new $(p,q)$ ( p , q ) -integral inequalities of Simpson’s second type for preinvex functions. Many results given in paper provide generalizations and extensions the previous research. Moreover, exampl...

In this paper, using a new identity, we study one of the famous Newton-Cotes three-point quadraturerules. More precisely Maclaurin’s quadrature rule, for which establish error estimate methodunder constraint that first derivatives belong to class preinvex functions. We also give someapplications special means as applications. believe studied inequality and resultsobtained in article will furthe...

assume that $mathbb{d}$ is the open unit disk. applying ozaki's conditions, we consider two classes of locally univalent, which denote by $mathcal{g}(alpha)$ and $mathcal{f}(mu)$ as follows begin{equation*}  mathcal{g}(alpha):=left{fin mathcal{a}:mathfrak{re}left( 1+frac{zf^{prime prime }(z)}{f^{prime }(z)}right)