Sufficient Conditions for Starlike Functions Associated with the Lemniscate of Bernoulli

Let -1\leq B<A\leq 1. Condition on \beta, is determined so that 1+\beta zp'(z)/p^k(z)\prec(1+Az)/(1+Bz)\;(-1<k\leq3) implies p(z)\prec \sqrt{1+z}. Similarly, condition on \beta is determined so that 1+\beta zp'(z)/p^n(z) or p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\;(n=0, 1, 2) implies p(z)\prec(1+Az)/(1+Bz) or \sqrt{1+z}. In addition to that condition on \beta is derived so that p(z)\prec(1+Az)/(1+Bz) when p(z)+\beta zp'(z)/p(z)\prec\sqrt{1+z}. Few more problems of the similar flavor are also considered

Certain Coefficient Problems of Se∗\mathcal{S}_{e}^{*} and Ce\mathcal{C}_{e}

In the present investigation, we consider the following classes of starlike and convex functions associated with an exponential function, respectively given by \begin{equation*} \mathcal{S}_{e}^{*}=\bigg\{f\in \mathcal {A}:\dfrac{zf'(z)}{f(z)}\prec e^{z}\bigg\}\quad \text{and}\quad \mathcal{C}_{e}=\bigg\{f\in \mathcal {A}:1+\dfrac{zf''(z)}{f'(z)}\prec e^{z}\bigg\}, \end{equation*} to establish certain coefficient related problems such as estimation of sharp bound of third Hankel determinant, bounds of sixth and seventh coefficients as well as the fourth order Hankel determinant

Subordination and Radius Problems for Certain Starlike Functions

We study the following class of starlike functions $$\mathcal{S}^*_{\wp}:=\left\{f\in\mathcal{A}: {zf'(z)}/{f(z)}\prec\ 1+ze^z=:\wp(z) \right\},$$ that are associated with the cardioid domain $\wp(\mathbb{D})$, by deriving certain convolution results, radius problems, majorization result, radius problems in terms of coefficients and differential subordination implications. Consequently, we establish some interesting generalizations of our results for the Ma-Minda class of starlike functions $\mathcal{S}^{*}(\psi)$. We also provide, the set of extremal functions maximizing $\Re\Phi\left(\log{(f(z)/z)}\right)$ or $\left|\Phi\left(\log{(f(z)/z)}\right)\right|$ for functions in $\mathcal{S}^{*}(\psi)$, where $\Phi$ is a non-constant entire function. Further T. H. MacGregor's result for the class $\mathcal{S}^{*}(\alpha)$ and $\mathcal{S}^*_{\wp}$ are obtained as special case to our result

Some Differential Subordination Results for a Class of Starlike Functions

In this paper, first order differential subordination implication results are derived for $\mathcal{S}^{*}_{\varrho},$ a subclass of starlike functions, defined as $$\mathcal{S}^{*}_{\varrho}=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\prec \varrho(z):=\cosh \sqrt{z} ,z\in\mathbb{D}\right\},$$ where we choose the branch of the square root function so that $\cosh\sqrt{z}=1+z/2!+z^{2}/{4!}+\cdots.$ Further we deduce Briot-Bouquet differential subordination results along with some examples.Comment: arXiv admin note: substantial text overlap with arXiv:2201.0581