In this paper, we utilize the $q$-Chu-Vandermonde formula to derive a novel expectation formula for the $q$-probability distribution $W(x,y;q)$, extending previously known results. Several applications are presented, including a broader generalization of the Andrews-Askey integral. Although fractional $q$-calculus is not directly employed in this work, its potential for future extensions is discussed, as non-integer order derivatives and integrals could offer deeper insights into $q$-series and probability distributions.

In recent years, many authors have studied and investigated majorization results for different subclasses of analytic functions. In this paper, we give some majorization results for certain non vanishing analytic functions, whose ratios are subordinated to different domains in the open unit disk.

Let $r\in \mathbb{C},$ $s\in [-1,0)$, $0\le \alpha <1.$ Then, $\mathcal{Q}\left[r,s,\alpha \right]$ stands for the set of analytic functions $q$ that is within the open unit disk $E$, with $q\left(0\right)=1,$ and satisfies the explicit representation

$$q\left(\zeta \right)=\frac{1+\left(\left(1-\alpha \right)r+\alpha s\right)\chi \left(\zeta \right)}{1+s\chi \left(\zeta \right)},$$

where $\chi \left(0\right)=0$ and $ \left \vert \chi \left(\zeta \right)\right \vert < 1. $Inthisarticle,wefindtheregionsofvariability$ W_{\lambda }\left(\zeta _{0}, r, s, \alpha \right) $for$ \int \limits_{0}^{z_{0}}q\left(\rho \right) d\rho \ $when$ q $rangesovertheclass$ \mathcal{Q}_{\lambda }\left[r, s, \alpha \right] $ defined as

$${\mathcal{Q}}_{\lambda }\left[r,s,\alpha \right]=\left\{q\in \mathcal{Q}\left[r,s,\alpha \right]:q^{\mathrm{\prime }}\left(0\right)=\left(\left(1-\alpha \right)\left(r-s\right)\right)\lambda \right\}$$

for any fixed ${\zeta }_0\in E$ and $\lambda \in \overline{E}$. As a corollary, the region of variability appears for the alternate sets of parameters as well.

Investigating the variability domain in the geometric function theory yields profound insights into the behavior of geometric functions, thereby facilitating the examination of extremal problems and the derivation of bounds and inequalities. While the previous literature has examined similar classes, our approach offers significant advantages through a more generalized framework. Our study considers normalized analytic functions with specific positivity conditions which involve complex parameters. This investigation extends a previous work by analyzing a broader set of non-vanishing analytic functions. Unlike earlier studies that focused on specific parameter values, our approach allows for wider applications across multiple subclasses through the incorporation of additional parameters. We aim to determine the variability domain for the logarithm of these functions at fixed points within the unit disk as the functions range over a particular class defined by the specific parameter constraints. This generalized approach unifies several known results and provides a comprehensive framework to solve previously intractable boundary problems in the geometric function theory.