For a function f starlike of order $$\alpha $$ , $$0\leqslant \alpha <1$$ non-constant polynomial Q degree n which is non-vanishing in the unit disc $${\mathbb {D}}$$ and $$\beta >0$$ we consider $$F:{\mathbb {D}}\rightarrow {\mathbb {C}}$$ defined by $$F(z)=f(z) (Q(z))^{\beta /n}$$ find largest value $$r\in (0,1]$$ such that $$r^{-1} F(rz)$$ lies various known subclasses functions as class $$\...

This paper introduces extensions H4,p and X8,p of Horn’s double hypergeometric function H4 Exton’s triple X8, taking into account recent Euler’s beta function, confluent function. Among the numerous extended functions, primary rationale for choosing X8 is their comparable extension type. Next, we present various integral representations Euler Laplace types, Mellin inverse transforms, Laguerre p...

Given $\beta\in(0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x) = ax + O\bigl(x\exp(-c\log^{\beta} x)\bigr)$ for some $a>0$ and $c>0$, prime function $\pi(x) \mathrm{Li}(x) \Omega\bigl(x\exp(-c'(\log x)^{\frac{\beta}{\beta+1}})\bigr)$ $c'>0$. This is done by generalizing construction Diamond, Montgomery, Vorhauer. serves as additional ...

The aim of this paper is twofold. Firstly, we will investigate the link between condition for functions $\phi(s)$ from $(\alpha, \beta)$-metrics Douglas type to be self-concordant and k-self concordant, other objective continue recently new introduced \beta)$-metric ([17]): $$ F(\alpha,\beta)=\frac{\beta^{2}}{\alpha}+\beta+a \alpha where $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ a Riemannian metric; $\b...