Toeplitz and Hankel determinants of logarithmic coefficients for r-valent q-starlike and r-valent q-convex functions

The aim of the present paper is to extend the notions of $q$-starlikeness and $q$-convexity to encompass multivalent $q$-starlikeness and multivalent $q$-convexity. We systematically introduce and examine subfamilies of $r$-valently holomorphic functions within the open unit disk $D$ by employing the fractional $q$-derivative operator, along with the principle of subordination between holomorphic functions.

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  The families we define in this paper constitute a generalization of numerous established classes available in existing literature.
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  We derive the Fekete-Szegö inequalities for these newly introduced families.
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  As a result, we apply these findings to establish bounds for the Toeplitz and Hankel determinants $T_{2,1}\left({\gamma }_u\right),T_{2,2}\left({\gamma }_u\right)$ and $H_{2,1}\left({\gamma }_u\right)$, defined as follows:$T_{2,1}\left({\gamma }_u\right)=\left|\begin{array}{cc}{\gamma }_1& {\gamma }_2\\ {\gamma }_2& {\gamma }_1\end{array}\right|,T_{2,2}\left({\gamma }_u\right)=\left|\begin{array}{cc}{\gamma }_2& {\gamma }_3\\ {\gamma }_3& {\gamma }_2\end{array}\right|\text{and}{H}_{2,1}\left({\gamma }_u\right)=\left|\begin{array}{cc}{\gamma }_1& {\gamma }_2\\ {\gamma }_2& {\gamma }_3\end{array}\right|$where ${\gamma }_1,{\gamma }_2$, and ${\gamma }_3$ denote the first, second, and third logarithmic coefficients of functions within the family of multivalent $q$-starlike and multivalent $q$-convex functions.