Starlike Functions Related to the Bell Numbers

Let $\mathcal{A}$ be a class of analytic functions f in the open unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ and normalized by the conditions $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$. Suppose $\mathcal{S}$ is a subclass of $\mathcal{A}$ consisting of univalent functions. An analytic function f is subordinate to g, written as $f\prec g$, if there exists an analytic function $w:\mathbb{D}\to \mathbb{D}$ with $\left|w\right(z\left)\right|\le \left|z\right|$ such that $f\left(z\right)=g\left(w\right(z\left)\right)(z\in \mathbb{D})$. Moreover, if g is univalent in $\mathbb{D}$, then the equivalent conditions for subordination can be written as $f\left(0\right)=g\left(0\right)\mathrm{and}f\left(\mathbb{D}\right)\subseteq g\left(\mathbb{D}\right)$. By imposing some geometric and analytic conditions over the functions in the class $\mathcal{S}$, many authors considered several subclasses of $\mathcal{S}$. Various subclasses of starlike and convex functions were studied in the literature, and they can be unified by considering an analytic univalent function $\phi$ with a positive real part in $\mathbb{D}$, symmetric about the real axis and starlike with respect to $\phi \left(0\right)=1$, and ${\phi }^{\prime }\left(0\right)>0$. Ma and Minda [1] studied the class

The class ${\mathcal{S}}^{*}\left(\phi \right)$ for various choice of the domain $\phi \left(\mathbb{D}\right)$ was considered in recent years. The class ${\mathcal{S}}^{*}[A,B]:={\mathcal{S}}^{*}((1+Az)/(1+Bz))(-1\le B<A\le 1)$ was introduced by Janowski [2]. For $0\le \alpha \le 1$, the class ${\mathcal{S}}^{*}\left(\alpha \right):={\mathcal{S}}^{*}[1-2\alpha ,-1]$ is the class of starlike functions of order $\alpha$. Uralegaddi et al. [3] defined the class

Several authors considered various special cases of the class of Janowski starlike functions by considering some specific functions, namely ${\phi }_q\left(z\right):=z+\sqrt{1+z^2}$, ${\phi }_0\left(z\right):=1+(z/k)((k+z)/(k-z))(k=\sqrt{2}+1)$, ${\phi }_s\left(z\right):=1+sinz$, and $G_{\alpha }\left(z\right):=1+z/(1-\alpha z^2)$. Some of those classes are: ${\mathcal{S}}_L^{*}:={\mathcal{S}}^{*}\left(\sqrt{1+z}\right)$ [4], ${\mathcal{S}}_q^{*}:={\mathcal{S}}^{*}\left({\phi }_q\left(z\right)\right)$ [5], ${\mathcal{S}}_e^{*}={\mathcal{S}}^{*}\left(e^z\right)$ [6], ${\mathcal{S}}_R^{*}={\mathcal{S}}^{*}\left({\phi }_0\right)$ [7], ${\mathcal{S}}_s^{*}={\mathcal{S}}^{*}\left({\phi }_s\right)$ [8]), ${\mathcal{BS}}^{*}\left(\alpha \right):={\mathcal{S}}^{*}\left(G_{\alpha }\left(z\right)\right),0\le \alpha <1$ [9,10]. For a brief survey on these classes, readers may refer to [11,12].

It should be noted that the special cases of $\phi$, mentioned above, are univalent in the unit disk. In 2011, Dziok et al. [13,14] considered $\phi$ to be a non-univalent function associated with the Fibonacci numbers, defined by which maps the unit disk $\mathbb{D}$ on to a shell-like domain in the right-half plane. Further, they defined the class ${\mathcal{S}}_F^{*}:=\left\{f\in \mathcal{A}:z{f}^{\prime }\left(z\right)/f\left(z\right)\prec \tilde{p}\left(z\right)\right\}$. The functions $f\in {\mathcal{S}}_F^{*}$ are starlike of order $\sqrt{5}/10$.

Motivated by the above defined classes, we consider a function associated with the Bell Numbers. For a fixed non-negative integer n, the Bell numbers $B_n$ count the possible disjoint partitions of a set with n elements into non-empty subsets or, equivalently, the number of equivalence relations on it. The Bell numbers $B_n$ satisfy a recurrence relation involving binomial coefficients $B_{n+1}={\sum }_{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k.$ Clearly $B_0=B_1=1,B_2=2,B_3=5,B_4=15,B_5=52$, and $B_6=203$. For more details, see [15,16,17,18,19,20,21]. Kumar et al. [22] considered the function which is starlike with respect to 1 and it’s coefficients generate the Bell numbers. Kumar et al. [22] defined the class ${\mathcal{S}}_B^{*}$ by ${\mathcal{S}}_B^{*}:={\mathcal{S}}^{*}\left(Q\right)$. From [1], note that the function $f\in {\mathcal{S}}_B^{*}$ if and only if there exists an analytic function $q$, satisfying $q\left(z\right)\prec Q\left(z\right)(z\in \mathbb{D})$, such that

The above representation shows that the functions in the class ${\mathcal{S}}_B^{*}$ can be seen as an integral transform $I\left(q\right(z\left)\right)$ of the function q with $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$. The reader may refer to the paper [23] and the references cited therein for integral transform related works. The authors in [22] determined sharp coefficient bounds on the six initial coefficients, Hankel determinant, and on the first three consecutive higher order Schwarzian derivatives for functions in the class ${\mathcal{S}}_B^{*}$.

Let $\mathcal{P}$ be the class of analytic functions $p:\mathbb{D}\to \mathbb{C}$ with $p\left(0\right)=1$ and $Rep\left(z\right)>0(z\in \mathbb{D})$. In 1989, Nunokawa et al. [24] showed that if $1+z{p}^{\prime }\left(z\right)\prec 1+z$, then $p\left(z\right)\prec 1+z$. In 2007, Ali et al. [25] computed the condition on $\beta$, in each case, for which $A,B,C,D,E,F\in [-1,1]$. Further, Ali et al. [26] determined some sufficient conditions for normalized analytic functions to lemniscate starlike functions. Recently, Kumar and Ravichandran [27] obtained sufficient conditions for first order differential subordinations so that the corresponding analytic function belongs to the class $\mathcal{P}$. In 2016, Tuneski [28] gave a criteria for analytic functions to be Janowski starlike. For more details, see [11,29,30,31,32,33].

Motivated by above works, in Section 2, using the theory of differential subordination developed by Miller and Mocanu, a sharp bound on parameter $\beta$ is determined in each case so that $p\left(z\right)\prec Q\left(z\right)$, whenever $1+\beta z{p}^{\prime }\left(z\right)/p^j\left(z\right)(j=0,1,2)$ is subordinate to the function ${\phi }_0\left(z\right)$ or $\sqrt{1+z}$ or $G_{\alpha }\left(z\right)$ or $(1+Az)/(1+Bz)$ or ${\phi }_s\left(z\right)$ or ${\phi }_q\left(z\right)$. Further, various sufficient conditions are obtained for $f\in \mathcal{A}$ to be in the class ${\mathcal{S}}_B^{*}$ as an application of these subordination results. In Section 3, ${\mathcal{S}}_B^{*}$-radius for the class of Janowski starlike functions and some other well-known classes of analytic functions are investigated.

Theorem 1 provides estimate on $\beta$ so that $p\left(z\right)\prec Q\left(z\right)$ holds, whenever $1+\beta z{p}^{\prime }\left(z\right)\prec {\phi }_0\left(z\right)$ or ${\phi }_s\left(z\right)$ or $\sqrt{1+z}$ or $G_{\alpha }\left(z\right)$ or $(1+Az)/(1+Bz)$ or ${\phi }_s\left(z\right)$ or ${\phi }_q\left(z\right)$ or $e^z$.

To prove our main results, we need the following lemma due to Miller and Mocanu:

Theorem 1 also provides the following various sufficient conditions for the normalized analytic functions f to be in the class ${\mathcal{S}}_B^{*}$.

Let function $f\in \mathcal{A}$ and set

If either of the following subordination holds

then $f\in {\mathcal{S}}_B^{*}$.

The next result gives sharp lower bound on $\beta$ such that subordination $p\prec Q$ holds, whenever $1+\beta z{p}^{\prime }\left(z\right)/p\left(z\right)\prec {\phi }_0\left(z\right)$ or ${\phi }_s\left(z\right)$ or $\sqrt{1+z}$ or $G_{\alpha }\left(z\right)$ or $(1+Az)/(1+Bz)$ or ${\phi }_s\left(z\right)$ or ${\phi }_q\left(z\right)$ or $e^z$.

Next, Theorem 2 also provides the following various sufficient conditions for the normalized analytic functions f to be in the class ${\mathcal{S}}_B^{*}$. Let the function $f\in \mathcal{A}$ and set

If either of the following subordination conditions are fulfilled:

then $f\in {\mathcal{S}}_B^{*}$.