Let \(\psi\) be univalent, we introduce a newly defined class of analytic functions, given by $$\begin{aligned} {\mathcal {F}}(\psi ):= \left\{ f\in {A}}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \psi (z),\ (0)=0 \right\} . \end{aligned}$$Note that functions in this may not univalent. Further, establish the growth theorem with some geometrical conditions on and obtain Koebe domain related shar...

We study integrals of the form $$\begin{aligned} \int _{-1}^1(C_n^{(\lambda )}(x))^2(1-x)^\alpha (1+x)^\beta {{\,\mathrm{\mathrm {d}}\,}}x, \end{aligned}$$ where $$C_n^{(\lambda )}$$ denotes Gegenbauer-polynomial index $$\lambda >0$$ and $$\alpha ,\beta >-1$$ . give exact formulas for their generating functions, obtain asymptotic as $$n\rightarrow \infty $$