In the current paper, we study a majorization issue for a general category ${\mathcal{S}}^{*}\left(\vartheta \right)$ of starlike functions, the region of which is often symmetric with respect to the real axis. For various special symmetric functions $\vartheta$ , corresponding consequences of the main result are also presented with some relevant connections of the outcomes rendered here with those obtained in recent research. Moreover, coefficient bounds for some majorized functions are estimated.

For a constant $alphain left(-frac{pi}{2},frac{pi}{2}right)$, we definea subclass of the spirallike functions, $SP_{p}(alpha)$, the setof all functions $fin mathcal{A}$[releft{e^{-ialpha}frac{zf'(z)}{f(z)}right}geqleft|frac{zf'(z)}{f(z)}-1right|.]In the present paper, we shall give the estimate of the norm of the pre-Schwarzian derivative $mathrm{T}_f=f''/f'$ where $|mathrm{T}_f|=sup_{zin Delta} (1-|z|^2)|mathrm{T}_f(z)|$ for the functions in $SP_{p}(alpha)$.

In this work, we explore the impact of integral operators such as the Libera and Alexander operators on specific families of analytic functions introduced in the literature and find some of their remarkable results. Using techniques from differential subordination and convolution theory, we establish results concerning the radius of convexity and convolution properties for these function classes. Additionally, we investigate how these integral operators influence the geometric properties of functions in $\mathcal{BS}\left(\mu \right)$ and $\mathcal{KS}\left(\mu \right)$, leading to new insights into their structural behavior.

In this note, we study the integral operators I-g(gamma,alpha) and J(g)(gamma,alpha) of an analytic function g on convex and starlike functions of a complex order. Then, we investigate the same operators on H-infinity and Besov spaces.

Many papers concern both the starlikeness and the convexity of Bernardi integral operator. Using the Nunokawa’s Lemma, we want to determine conditions for the strong starlikeness of the Bernardi transform of normalized analytic functions g, such that $|arg\left\{g^{\prime }\left(z\right)\right\}| <{\displaystyle \frac{\alpha \pi }{2}}$ in the open unit disk $\Delta$ where $0<\alpha <2$. Our results include the results of Mocanu, Nunokawa and others on the Libera transform.