Related papers: Sufficient conditions for strong starlikeness

Let A,B,D,E belong to [-1, 1] and let p(z) be an analytic function with fixed initial coefficient defined in the open unit disk. Conditions on A,B,D,and E are determined so that 1+{\alpha}zp'(z) being subordinated to (1+Dz)/(1+Ez) implies…

The conditions on $A$, $B$, $\beta$ and $\gamma$ are obtained for an analytic function $p$ defined on the open unit disc $\mathbb{D}$ and normalized by $p(0)=1$ to be subordinate to $(1+Az)/(1+Bz)$, $-1\leq B<A \leq 1$ when $p(z)+…

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…

Sharp estimates on $\beta$ are determined so that an analytic function $p$ defined on the open unit disk in the complex plane normalized by $p(0)=1$ is subordinate to some well known starlike functions with positive real part whenever…

For functions $f(z)=z^p+a_{n+1}z^{p+1}+...$ defined on the open unit disk, the condition $\Re (f'(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity…

Sufficient conditions on $A$, $B$, $p$, $b$ and $c$ are determined that will ensure the generalized Bessel functions ${u}_{p,b,c}$ satisfies the subordination ${u}_{p,b,c}(z) \prec (1+Az)/ (1+Bz)$. In particular this gives conditions for…

The association of subordination and special functions is used to find sharp estimates on the parameter $\beta$ such that the analytic function $p(z)$ is subordinate to certain functions having positive real part whenever $p(z)+\beta z…

The known Ozaki's condition says that $\mathfrak{Re}\left\{f^{(p)}(z)\right\}>0$ for $|z|<1$ implies that $f(z)=z^p+a_{p+1}z^{p+1}+\cdots$ is at most $p$-valent in $\mathbb D$. In this paper prove an extension of Ozaki's condition. Also, we…

It is well-known that the condition ${\operatorname{Re}} \left[1+\frac{zf''(z)}{f'(z)}\right]>0$, $z\in{\mathbb D}$, implies that $f$ is starlike function (i.e. convexity implies starlikeness). If the previous condition is not satisfied for…

Let $f$ and $g$ be analytic functions on the open unit disk of the complex plane with $f/g$ belonging to the class $\mathcal{P} $ of functions with positive real part consisting of functions $p$ with $p(0)=1$ and $\operatorname{Re} p(z)>0$…

For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq \beta, sharp bound on $\beta$ is determined so that $f$ is either starlike or convex of order $\alpha$. Several other…

Sufficient conditions on associated parameters $p,b$ and $c$ are obtained so that the generalized and \textquotedblleft{normalized}\textquotedblright{} Bessel function $u_p(z)=u_{p,b,c}(z)$ satisfies $|(1+(zu''_p(z)/u'_p(z)))^2-1|<1$ or…

Let $\mathcal{A}$ consists of analytic functions $f:\mathbb{D}\to\mathbb{C}$ satisfying $f(0)=f'(0)-1=0$. Let $\mathcal{S}^*_{Ne}$ be the recently introduced Ma-Minda type functions family associated with the $2$-cusped kidney-shaped {\it…

The main object of this paper is to find necessary and sufficient conditions for generalized Bessel functions of first kind $zu_{p}(z)$ to be in the classes $\mathcal{SP}_{p}(\alpha ,\beta )$ and $\mathcal{UCSP}(\alpha ,\beta )$ of…

Given a domain $\Omega$ in the complex plane $\mathbb{C}$ and a univalent function $q$ defined in an open unit disk $\mathbb{D}$ with nice boundary behaviour, Miller and Mocanu studied the class of admissible functions $\Psi(\Omega,q)$ so…

For $0\leq \alpha <1$, the sharp radii of starlikeness and convexity of order $\alpha$ for functions of the form $f(z)=z+a_2z^2+a_3z^3+...$ whose Taylor coefficients $a_n$ satisfy the conditions $|a_2|=2b$, $0\leq b\leq 1$, and $|a_n|\leq n…

We are interested in finding the sufficient conditions on $A$, $B$, $\lambda$, $b$ and $c$ which ensure that the generalized Bessel functions ${u}_{\lambda}:={u}_{\lambda,b,c}$ satisfies the subordination ${u}_{\lambda}(z) \prec (1+Az)/…

There are many results for sufficient conditions of functions f(z) which are analytic in the open unit disc U to be starlike and convex in U. The object of the present paper is to derive some interesting sufficient conditions for f(z) to be…

Marx and Strohh\"acker showed around in 1933 that $f(z)/z$ is subordinate to $1/(1-z)$ for a normalized convex function $f$ on the unit disk $|z|<1.$ Brickman, Hallenbeck, MacGregor and Wilken proved in 1973 further that $f(z)/z$ is…

For an analytic function $f$ on the unit disk $\mathbb{D}=\{z:|z|<1\}$ satisfying $f(0)=0=f'(0)-1,$ we obtain sufficient conditions so that $f$ satisfies $|(zf'(z)/f(z))^2-1|<1.$ The technique of differential subordination of first or…