Skip to main content

Differential inequalities for spirallike and strongly starlike functions

Abstract

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that \(p(0)=1\) to satisfy \(\operatorname{Re}\{ {\mathrm{e}}^{{\mathrm{i}}\beta } p(z) \} > \gamma \) or \(| \arg \{p(z)-\gamma \} |<\delta \) for all \(z\in \mathbb{D}\), where \(\beta \in (-\pi /2,\pi /2)\), \(\gamma \in [0,\cos \beta )\), \(\delta \in (0,1]\) and \(\mathbb{D}:=\{z\in \mathbb{C}:|z|<1 \}\). The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in \(\mathbb{D}\).

Introduction and definitions

For real numbers β, γ, and δ satisfying \(-\pi /2 < \beta <\pi /2\), \(0 \leq \gamma < \cos \beta \), and \(0 < \delta \leq 1\), define two domains \(\Omega _{\gamma }(\beta )\) and \(\Lambda _{\gamma }(\delta )\) in \(\mathbb{C}\) by

$$ \Omega _{\gamma }(\beta ) = \bigl\{ w\in \mathbb{C}: \operatorname{Re} \bigl( {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } w \bigr) > \gamma \bigr\} $$

and

$$ \Lambda _{\gamma }(\delta ) = \biggl\{ w\in \mathbb{C}: \bigl\vert \arg (w- \gamma ) \bigr\vert < \frac{\pi }{2}\delta \biggr\} , $$

respectively. Then it clearly holds that

$$ \Omega _{\gamma }(\beta ) \cap \Omega _{\gamma }(- \beta ) \subset \Lambda _{\gamma } \biggl( 1-\frac{2}{\pi }\beta \biggr). $$
(1.1)

Let \(\mathbb{D}:=\{ z\in \mathbb{C}: |z|<1 \}\) be the open unit disk. Let \({\mathcal{H}}\) be the class of analytic functions in \(\mathbb{D}\), and let \({\mathcal{H}}_{1}\) be the class of functions \(p \in {\mathcal{H}}\) with \(p(0)=1\). We introduce two subfamilies \({\mathcal{P}}_{\gamma }(\beta )\) and \({\mathcal{Q}}_{\gamma }(\delta )\) of \({\mathcal{H}}_{1}\) defined as follows:

$$ {\mathcal{P}}_{\gamma }(\beta ) = \bigl\{ p \in {\mathcal{H}}_{1}: p(z) \in \Omega _{\gamma }(\beta ) \text{ for all } z\in \mathbb{D} \bigr\} $$

and

$$ {\mathcal{Q}}_{\gamma }(\delta ) = \bigl\{ p \in {\mathcal{H}}_{1}: p(z) \in \Lambda _{\gamma }(\delta ) \text{ for all } z\in \mathbb{D} \bigr\} . $$

A function p in \({\mathcal{P}}_{\gamma }(0)\) is said to be a Carathéodory function of order γ in \(\mathbb{D}\). In particular, \({\mathcal{P}}_{0}(0) \equiv {\mathcal{P}}\) is the well-known class of Carathéodory functions. Also, a function p in \({\mathcal{P}}_{0}(\beta )\) is said to be a tilted Carathéodory function by angle β [27]. We note that

$$ {\mathcal{P}}_{\gamma }(\beta ) \cap {\mathcal{P}}_{\gamma }(- \beta ) \subset {\mathcal{Q}}_{\gamma } \biggl( 1-\frac{2}{\pi } \beta \biggr) $$

holds, by (1.1).

Let \({\mathcal{A}}\) denote the class of functions f in \({\mathcal{H}}\) normalized by \(f(0)=0=f'(0)-1\). And let \({\mathcal{S}}\) be the subclass of \({\mathcal{A}}\) consisting of all univalent functions. Further we denote by \({\mathcal{S}}_{\gamma }^{*}(\beta )\) and \({\mathcal{SS}}_{\gamma }^{*}(\delta )\) the subclass of \({\mathcal{A}}\) consisting of β-spirallike functions of order γ [8, II, p. 89] (see also [16, 24]) and strongly starlike functions of order δ and type γ [9]. That is, a function \(f \in {\mathcal{A}}\) belongs to the class \({\mathcal{S}}_{\gamma }^{*}(\beta )\) if f satisfies

$$ \operatorname{Re} \biggl\{ {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } \frac{zf'(z)}{f(z)} \biggr\} > \gamma , \quad z\in \mathbb{D}, $$

and belongs to the class \({\mathcal{SS}}_{\gamma }^{*}(\delta )\) when f satisfies

$$ \biggl\vert \arg \biggl( \frac{zf'(z)}{f(z)} - \gamma \biggr) \biggr\vert < \frac{\pi }{2}\delta , \quad z\in \mathbb{D}. $$

Thus we have

$$ f \in {\mathcal{S}}_{\gamma }^{*}(\beta ) \quad \Longleftrightarrow\quad J_{f} \in {\mathcal{P}}_{\gamma }(\beta ) $$

and

$$ f \in {\mathcal{SS}}_{\gamma }^{*}(\delta ) \quad \Longleftrightarrow\quad J_{f} \in {\mathcal{Q}}_{\gamma }(\delta ), $$

where \(J_{f}(z):=zf'(z)/f(z)\), \(z\in \mathbb{D}\). Note that \({\mathcal{S}}_{\gamma }^{*}(0) \equiv {\mathcal{S}}^{*}(\gamma )\) is the class of starlike functions of order γ, and \({\mathcal{S}}_{0}^{*}(\beta ) \equiv {\mathcal{SP}}(\beta )\) is the class of β-spirallike functions. It is well known [24] (or [8, Vol. I, p. 149]) that \({\mathcal{S}}^{*}(\gamma )\) and \({\mathcal{SP}}(\beta )\) are the subclasses of \({\mathcal{S}}\). See [7, 12, 28] for sufficient conditions for spirallike functions. We also note that \({\mathcal{SS}}_{\gamma }^{*}(\delta ) \subset {\mathcal{S}}_{\gamma }^{*}(0) \subset {\mathcal{S}}\). Especially, \({\mathcal{SS}}_{0}^{*}(\delta ) \equiv {\mathcal{SS}}^{*}(\delta )\) which is the class of strongly starlike functions of order δ [4, 25]. Refer to [5, 6, 11, 13, 14, 1720, 23, 26] for various sufficient conditions for strongly starlike functions.

In the present paper we investigate new sufficient conditions for functions in \({\mathcal{P}}_{\gamma }(\beta )\) or \({\mathcal{Q}}_{\gamma }(\delta )\). As direct consequences of these results, we will obtain several sufficient conditions for spirallike functions or strongly starlike functions in \(\mathbb{D}\).

For analytic functions f and g, we say that f is subordinate to g, denoted by \(f\prec g\), if there is an analytic function \(\omega :\mathbb{D}\rightarrow \mathbb{D}\) with \(|\omega (z)|\leq |z|\) such that \(f(z)=g(\omega (z))\). Further, if g is univalent, then the definition of subordination \(f\prec g\) simplifies to the conditions \(f(0)=g(0)\) and \(f(\mathbb{D})\subseteq g(\mathbb{D})\) (see [21, p. 36]).

Let \(\overline{\mathbb{D}}=\{ z\in \mathbb{C}: |z| \leq 1 \}\) and \(\partial \mathbb{D}=\{ z\in \mathbb{C}: |z| = 1 \}\) be the closure and boundary of \(\mathbb{D}\), respectively. We denote by \({\mathcal{R}}\) the class of functions q that are analytic and injective on \(\overline{\mathbb{D}}\setminus E(q)\), where

$$ E(q) = \Bigl\{ \zeta : \zeta \in \partial \mathbb{D} \text{ and } \lim_{z\rightarrow \zeta }q(z)=\infty \Bigr\} , $$

and are such that

$$ q'(\zeta )\neq0 \quad \bigl(\zeta \in \partial \mathbb{D} \setminus E(q) \bigr). $$

Furthermore, let the subclass of \({\mathcal{R}}\) for which \(q(0)=a\) be denoted by \({\mathcal{R}}(a)\). We recall the following lemma which will be used for our results.

Lemma 1.1

([15, p. 24])

Let \(q \in {\mathcal{R}}(a)\) and let

$$ p(z) = a + a_{n}z^{n} + \cdots \quad (n\geq 1) $$

be an analytic function in \(\mathbb{D}\) with \(p(0) = a\). If p is not subordinate to q, then there exist points \(z_{0} \in \mathbb{D}\) and \(\zeta _{0} \in \partial \mathbb{D} \setminus E(q)\) for which

  1. (i)

    \(p(z_{0}) = q(\zeta _{0})\);

  2. (ii)

    \(z_{0}p'(z_{0}) = m\zeta _{0}q'(\zeta _{0})\) \((m\geq n\geq 1)\).

Main results

Throughout this section, let β and γ be real numbers such that \(-\pi /2 < \beta < \pi /2\) and \(0 \leq \gamma < \cos \beta \) unless we mention it. We define a function \(\varphi _{\beta ,\gamma } :\mathbb{D} \rightarrow \mathbb{C}\) by

$$ \varphi _{\beta ,\gamma }(z) = \frac{ 1+ ({\mathrm{{e}}}^{{\mathrm{{i}}}\beta }-2\gamma ) {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } z }{ 1 - z}. $$
(2.1)

Then it is easy to check that the bilinear function \(\varphi _{\beta ,\gamma }\) maps the unit disk \(\mathbb{D}\) onto the half-plane \(\Omega _{\gamma }(\beta )\). By using the function \(\varphi _{\beta ,\gamma }\) we obtain the following results.

Theorem 2.1

Let \(\alpha \in \mathbb{C}\) with \(\operatorname{Re}(\alpha ) \geq 0\). If \(p \in {\mathcal{H}}_{1}\) satisfies

$$ \biggl\vert p(z) + \alpha \frac{zp'(z)}{p(z)} - 1 \biggr\vert < (\cos \beta - \gamma ) \biggl( 1+\frac{1}{2}\operatorname{Re}( \alpha ) \biggr) \bigl\vert p(z) \bigr\vert , \quad z\in \mathbb{D}, $$
(2.2)

then \(1/p \in {\mathcal{P}}_{\gamma }(-\beta )\). That is, \(\operatorname{Re}\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }/p(z) \} > \gamma \) for all \(z\in \mathbb{D}\).

Proof

Let us define functions q and \(h:\mathbb{D}\rightarrow \mathbb{C}\) by

$$ q(z) = \frac{{\mathrm{{e}}}^{{\mathrm{{i}}}\beta }}{p(z)} $$
(2.3)

and

$$ h(z) = {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }\varphi _{-\beta ,\gamma }(z) = \frac{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }+({\mathrm{{e}}}^{-{\mathrm{{i}}}\beta }-2\gamma )z }{ 1-z }, $$
(2.4)

where \(\varphi _{\beta ,\gamma }\) is the function defined by (2.1). Then the functions q and h are analytic in \(\mathbb{D}\) with

$$ q(0)=h(0)={\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \in \mathbb{C} \quad \text{and} \quad h(\mathbb{D}) = \bigl\{ w\in \mathbb{C}:\operatorname{Re} \{ w \} >\gamma \bigr\} . $$

Suppose now that q is not subordinate to h. Then, by Lemma 1.1, there exist points \(z_{0}\in \mathbb{D}\) and \(\zeta _{0} \in \partial \mathbb{D} \setminus \{ 1 \} \) such that

$$ q(z_{0})=h(\zeta _{0})=\gamma + { \mathrm{{i}}}\rho \quad (\rho \in \mathbb{R}) \quad \text{and} \quad z_{0}q'(z_{0}) = m \zeta _{0} h'(\zeta _{0}) = m \sigma \quad (m\geq 1), $$
(2.5)

where

$$ \sigma = \frac{ -\rho ^{2} +2\rho \sin \beta +2\gamma \cos \beta -1-\gamma ^{2} }{ 2(\cos \beta -\gamma ) }. $$
(2.6)

Since \(\gamma < \cos \beta \), we get \(\sigma <0\). Indeed, we have

$$ 2(\cos \beta -\gamma )\sigma = -(\rho -\sin \beta )^{2} -(\cos \beta - \gamma )^{2} \leq -(\cos \beta -\gamma )^{2}, $$

which implies that

$$ \sigma \leq -\frac{1}{2}(\cos \beta -\gamma ) < 0. $$

Using (2.3) and (2.5), we have

$$ \begin{aligned}[b] \biggl\vert \frac{ p(z_{0}) + \alpha \frac{z_{0} p'(z_{0}) }{p(z_{0})} -1 }{ p(z_{0}) } \biggr\vert &= \bigl\vert \alpha z_{0} q'(z_{0}) + q(z_{0}) -{\mathrm{{e}}}^{{ \mathrm{{i}}}\beta } \bigr\vert \\ &= \bigl\vert \alpha m \sigma + \gamma +{\mathrm{{i}}}\rho -{ \mathrm{{e}}}^{{\mathrm{{i}}} \beta } \bigr\vert . \end{aligned} $$
(2.7)

Let \(\alpha =\alpha _{1}+{\mathrm{{i}}}\alpha _{2}\) with \(\alpha _{1} \geq 0\) and \(\alpha _{2}\in \mathbb{R}\). Then we have

$$ \begin{aligned}[b] & \bigl\vert \alpha m \sigma + \gamma +{\mathrm{{i}}}\rho -{\mathrm{{e}}}^{{ \mathrm{{i}}}\beta } \bigr\vert ^{2} \\ &\quad = \bigl( \alpha m \sigma + \gamma +{\mathrm{{i}}}\rho -{\mathrm{{e}}}^{{\mathrm{{i}}} \beta } \bigr) \bigl( \overline{\alpha } m \sigma + \gamma -{\mathrm{{i}}}\rho -{ \mathrm{{e}}}^{-{ \mathrm{{i}}}\beta } \bigr) \\ &\quad = \vert \alpha \vert ^{2} m^{2} \sigma ^{2} +(\gamma -\cos \beta )^{2} +2 \alpha _{1} m\sigma (\gamma -\cos \beta ) + \kappa , \end{aligned} $$
(2.8)

where

$$ \kappa = (\rho -\sin \beta )^{2} +2\alpha _{2} m\sigma ( \rho -\sin \beta ). $$

Furthermore it is easy to see that

$$ \kappa = ( \rho -\sin \beta + m \alpha _{2} \sigma )^{2} - (m \alpha _{2} \sigma )^{2} \geq - (m \alpha _{2} \sigma )^{2}. $$

Since \(m \geq 1\), from (2.8), we have

$$ \begin{aligned}[b] & \bigl\vert \alpha m \sigma + \gamma +{\mathrm{{i}}}\rho -{\mathrm{{e}}}^{{ \mathrm{{i}}}\beta } \bigr\vert ^{2} \\ &\quad \geq \vert \alpha \vert ^{2} m^{2} \sigma ^{2} +(\gamma -\cos \beta )^{2} +2 \alpha _{1} m\sigma (\gamma -\cos \beta ) - \alpha _{2}^{2} m^{2} \sigma ^{2} \\ &\quad = \alpha _{1}^{2} m^{2} \sigma ^{2} +(\gamma -\cos \beta )^{2} +2 \alpha _{1} m\sigma (\gamma -\cos \beta ) \\ &\quad \geq \alpha _{1}^{2} \sigma ^{2} +(\gamma -\cos \beta )^{2} +2 \alpha _{1} \sigma (\gamma -\cos \beta ) \\ &\quad = [ \alpha _{1} \sigma +\gamma - \cos \beta ]^{2}. \end{aligned} $$
(2.9)

Since \(\sigma <0\), \(\alpha _{1}\geq 0\), and \(\cos \beta >\gamma \), inequality (2.9) implies

$$ \bigl\vert \alpha m \sigma + \gamma +{\mathrm{{i}}}\rho -{ \mathrm{{e}}}^{{\mathrm{{i}}}\beta } \bigr\vert \geq -\alpha _{1}\sigma + \cos \beta - \gamma . $$
(2.10)

Furthermore, since \(\sigma \leq -(\cos \beta -\gamma )/2\), we have

$$ -\alpha _{1}\sigma +\cos \beta -\gamma \geq (\cos \beta -\gamma ) \biggl( 1+\frac{1}{2}\alpha _{1} \biggr). $$
(2.11)

Finally, from (2.7), (2.10), and (2.11), we obtain

$$ \biggl\vert p(z_{0}) + \alpha \frac{z_{0}p'(z_{0})}{p(z_{0})} - 1 \biggr\vert \geq (\cos \beta -\gamma ) \biggl( 1+\frac{1}{2}\alpha _{1} \biggr) \bigl\vert p(z_{0}) \bigr\vert . $$

This inequality contradicts hypothesis (2.2). Therefore, we obtain \(q \prec h\) in \(\mathbb{D}\) and the inequality \(\operatorname{Re}\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }/p(z) \} > \gamma \) holds for all \(z\in \mathbb{D}\). □

We remark that the hypothesis in Theorem 2.1 implies also \(1/p \in {\mathcal{P}}_{\gamma }(\beta )\). And we also remark that Theorem 2.1 reduces the result [13] when \(\alpha =1\).

By the above remark, taking \(\gamma =1/2\) in Theorem 2.1 gives the following corollary.

Corollary 2.1

Let α and \(\beta \in \mathbb{R}\) with \(\alpha \geq 0\) and \(\beta \in [0,\pi /3)\). If \(p \in {\mathcal{H}}_{1}\) satisfies (2.2) with \(\gamma =1/2\), then \(p(\mathbb{D}) \subset \Xi _{\beta }\), where

$$ \Xi _{\beta }= \bigl\{ w\in \mathbb{C}: \bigl\vert { \mathrm{{e}}}^{-{\mathrm{{i}}}\beta } w -1 \bigr\vert < 1 \textit{ and } \bigl\vert {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } w -1 \bigr\vert < 1 \bigr\} , $$

and we have \(\operatorname{Re}\{ p(z) \} > 0\) for all \(z\in \mathbb{D}\). Furthermore, if \(\beta \neq0\), then \(|\arg \{ p(z) \}| < \cot \beta \) for all \(z\in \mathbb{D}\).

Taking \(p(z)=zf'(z)/f(z)\), \(f\in {\mathcal{A}}\), in Corollary 2.1 gives the following result.

Corollary 2.2

Let \(\alpha \in \mathbb{R}\) with \(\alpha \geq 0\). If \(\beta \in (0,\pi /3)\) and \(f\in {\mathcal{A}}\) satisfies

$$ \biggl\vert (1-\alpha ) \frac{zf'(z)}{f(z)} + \alpha \biggl( 1+ \frac{zf''(z)}{f'(z)} \biggr) -1 \biggr\vert < \sqrt{\alpha +1} \biggl( \cos \beta -\frac{1}{2} \biggr) \biggl\vert \frac{zf'(z)}{f(z)} \biggr\vert , \quad z\in \mathbb{D}, $$
(2.12)

then \(f \in {\mathcal{SS}}_{0}^{*}(\cot \beta )\), i.e., f is strongly starlike of order \(2(\cot \beta )/\pi \) in \(\mathbb{D}\). If \(f\in {\mathcal{A}}\) satisfies (2.12) with \(\beta =0\), then \(f \in {\mathcal{S}}_{0}^{*}(0)\), i.e., f is a starlike function in \(\mathbb{D}\).

Example 2.1

Let \(a\in \mathbb{C}\) be given, and let \(f_{a}(z)=z/(1-az)\), \(z\in \mathbb{D}\). Then a computation shows that

$$ \frac{1}{2} \biggl\vert \frac{zf'(z)}{f(z)} + \frac{zf''(z)}{f'(z)} -1 \biggr\vert = \frac{3 \vert a \vert \vert z \vert }{2 \vert 1-az \vert } < \frac{3 \vert a \vert }{2 \vert 1-az \vert } = \frac{3 \vert a \vert }{2} \biggl\vert \frac{zf'(z)}{f(z)} \biggr\vert , \quad z\in \mathbb{D}. $$

Hence if

$$ \vert a \vert < \frac{ \sqrt{6} }{3} \biggl( \cos \beta - \frac{1}{2} \biggr), $$
(2.13)

then inequality (2.12) with \(\alpha =1/2\) holds. Thus, by Corollary 2.2 with \(\alpha =1/2\), we conclude that \(f_{a}\) is strongly starlike of order \(2(\cot \beta )/\pi \) in \(\mathbb{D}\) provided inequality (2.13) holds.

Example 2.2

Let \(g_{a}(z)=z/(1-az)^{2}\), \(z\in \mathbb{D}\), with \(a\in \mathbb{C}\). Then a similar computation with Example 2.1 and Corollary 2.2 gives that if \(a\in \mathbb{C}\) satisfies

$$ \frac{2 \vert a \vert (3+2 \vert a \vert )}{ (1- \vert a \vert )^{2} } \leq \frac{ \sqrt{6} }{2} \biggl( \cos \beta - \frac{1}{2} \biggr), $$

then \(g_{a}\) is strongly starlike of order \(2(\cot \beta )/\pi \) in \(\mathbb{D}\).

Theorem 2.2

Let \(\alpha \in \mathbb{R}\) with \(\alpha \geq 0\). Assume that

$$ (2\lambda +\gamma ) \vert \sin \beta \vert < 2\sqrt{\Delta }, $$
(2.14)

where

$$ \Delta =\lambda (\lambda + \cos \beta ) \bigl(-2\gamma \cos \beta +1 + \gamma ^{2} \bigr), $$
(2.15)

with

$$ \lambda = \frac{\alpha }{ 2(\cos \beta -\gamma ) } \geq 0. $$

Let \(p \in {\mathcal{H}}_{1}\) with \(\gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } \notin p(\mathbb{D})\). If

$$ \biggl\vert \operatorname{Im} \biggl\{ p(z) + \frac{\alpha zp'(z)}{p(z) - \gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } } + { \mathrm{{i}}}(2 \lambda +\gamma )\sin \beta \biggr\} \biggr\vert < 2\sqrt{ \Delta }, \quad z\in \mathbb{D}, $$
(2.16)

then \(p \in {\mathcal{P}}_{\gamma }(-\beta )\). That is, \(\operatorname{Re}\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }p(z) \} > \gamma \) for all \(z\in \mathbb{D}\).

Proof

We first note that, since \(p(0)=1\), (2.14) implies that inequality (2.16) is well-defined. Next we define functions q and h by

$$ q(z) = {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }p(z) $$
(2.17)

and (2.4), respectively. If q is not subordinate to h, then there exist points \(z_{0} \in \mathbb{D}\) and \(\zeta _{0} \in \partial \mathbb{D} \setminus \{ 1 \}\) satisfying (2.5) with \(\rho \in \mathbb{R}\). We note that \(\rho \neq0\). Indeed, if \(\rho =0\), then \({\mathrm{{e}}}^{{\mathrm{{i}}}\beta }p(z_{0}) = q(z_{0}) = \gamma \). Therefore we have \(p(z_{0}) = \gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta }\), which contradicts the condition \(\gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } \notin p(\mathbb{D})\).

Simple computations give

$$ \begin{aligned} &p(z_{0}) + \frac{ \alpha z_{0} p'(z_{0}) }{ p(z_{0}) - \gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } } \\ &\quad = {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta }q(z_{0}) + \frac{ \alpha z_{0} q'(z_{0}) }{ q(z_{0}) -\gamma } \\ &\quad = \gamma \cos \beta + \rho \sin \beta +{\mathrm{{i}}} \biggl( \rho \cos \beta - \gamma \sin \beta - \frac{\alpha m \sigma }{\rho } \biggr), \end{aligned} $$

where σ is given by (2.6). Therefore we get

$$ \begin{aligned}[b] &\operatorname{Im} \biggl\{ p(z_{0}) + \frac{\alpha z_{0}p'(z_{0})}{p(z_{0}) - \gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } } + {\mathrm{{i}}}(2 \lambda +\gamma )\sin \beta \biggr\} \\ &\quad = \rho \cos \beta - \frac{\alpha m \sigma }{\rho } + 2 \lambda \sin \beta \\ &\quad = \rho \cos \beta +m \lambda \biggl[ \rho - 2\sin \beta + \frac{-2\gamma \cos \beta +1+\gamma ^{2}}{\rho } \biggr] +2 \lambda \sin \beta . \end{aligned} $$
(2.18)

Assume that \(\rho >0\), and put

$$ \tilde{ \lambda } = \rho - 2\sin \beta + \frac{-2\gamma \cos \beta +1+\gamma ^{2}}{\rho }. $$

Note that

$$ -2\gamma \cos \beta +1+\gamma ^{2} = (\gamma -\cos \beta )^{2} +\sin ^{2} \beta \geq 0 $$

and

$$ -2\gamma \cos \beta +1+\gamma ^{2} - \sin ^{2}\beta = ( \cos \beta - \gamma )^{2} \geq 0. $$

Since \(\rho >0\), these inequalities yield that

$$ \tilde{ \lambda } \geq 2\sqrt{-2\gamma \cos \beta +1+\gamma ^{2}} -2 \sin \beta \geq 0. $$

Therefore, since \(m \geq 1\) and \(\lambda \geq 0\), from (2.18), we obtain

$$ \begin{aligned}[b] &\operatorname{Im} \biggl\{ p(z_{0}) + \frac{\alpha z_{0}p'(z_{0})}{p(z_{0}) - \gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } } + {\mathrm{{i}}}(2 \lambda +\gamma )\sin \beta \biggr\} \\ &\quad \geq \rho \cos \beta + \lambda \tilde{ \lambda } +2 \lambda \sin \beta \\ &\quad = (\cos \beta + \lambda )\rho + \frac{ \lambda (-2\gamma \cos \beta +1+\gamma ^{2})}{\rho } \\ &\quad \geq 2\sqrt{\Delta }, \end{aligned} $$
(2.19)

where Δ is given by (2.15). This contradicts condition (2.16).

Now assume that \(\rho <0\). From (2.18), we have

$$ \begin{aligned} &\operatorname{Im} \biggl\{ p(z_{0}) + \frac{\alpha z_{0}p'(z_{0})}{p(z_{0}) - \gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } } + {\mathrm{{i}}}(2 \lambda +\gamma )\sin \beta \biggr\} \\ &\quad = - \biggl[ \tilde{\rho } \cos \beta +m \lambda \biggl( \tilde{\rho } +2 \sin \beta + \frac{-2\gamma \cos \beta +1+\gamma ^{2}}{\tilde{\rho }} \biggr) \biggr] + 2 \lambda \sin \beta , \end{aligned} $$

where \(\tilde{\rho } = -\rho >0\). A similar calculation with (2.19) gives us to get

$$ \operatorname{Im} \biggl\{ p(z_{0}) + \frac{\alpha z_{0}p'(z_{0})}{p(z_{0}) - \gamma {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } } + { \mathrm{{i}}}(2 \lambda +\gamma )\sin \beta \biggr\} \leq -2\sqrt{ \Delta }, $$

where Δ is given by (2.15). This also contradicts condition (2.16). Therefore we get \(q \prec h\) in \(\mathbb{D}\), and the inequality \(\operatorname{Re}\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }p(z) \} > \gamma \), \(z\in \mathbb{D}\), follows. □

We remark that Theorem 2.2 reduces the result [13] when \(\alpha =1\).

Theorem 2.3

Let \(\alpha \in \mathbb{C}\) with \(\operatorname{Re}(\alpha ) \geq 0\). Assume that \(\Psi (\alpha ,\beta ,\gamma ) < \cos \beta \), where

$$ \Psi (\alpha ,\beta ,\gamma ) = \frac{ \sin ^{2}\beta (\gamma +s)^{2} }{ \cos \beta +s } + \gamma ^{2} \cos \beta +s \bigl(2\gamma \cos \beta -1-\gamma ^{2} \bigr), $$
(2.20)

with

$$ s= \frac{ \operatorname{Re}(\alpha ) }{ 2(\cos \beta -\gamma ) }. $$

If \(p \in {\mathcal{H}}_{1}\) satisfies

$$ \operatorname{Re} \bigl\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \bigl[ \bigl(p(z) \bigr)^{2} + \alpha z p'(z) \bigr] \bigr\} > \Psi (\alpha ,\beta ,\gamma ), \quad z \in \mathbb{D}, $$
(2.21)

then \(\operatorname{Re}\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }p(z) \} > \gamma \) for all \(z\in \mathbb{D}\).

Proof

We first note that, since \(p(0)=1\), the hypothesis \(\Psi (\alpha ,\beta ,\gamma ) < \cos \beta \) implies that inequality (2.21) is well defined. Now we define the functions q and h by (2.17) and (2.4), respectively. If q is not subordinate to h, then there exist points \(z_{0} \in \mathbb{D}\) and \(\zeta _{0} \in \partial \mathbb{D} \setminus \{ 1 \}\) satisfying (2.5) with \(\rho \in \mathbb{R}\).

Put \(\alpha = \alpha _{1} + {\mathrm{{i}}}\alpha _{2}\) with \(\alpha _{1} \geq 0\) and \(\alpha _{2}\in \mathbb{R}\). By (2.17) and (2.5), we obtain

$$ \begin{aligned} & {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \bigl[ \bigl(p(z_{0}) \bigr)^{2} + \alpha z_{0} p'(z_{0}) \bigr] \\ &\quad = {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \bigl[ {\mathrm{{e}}}^{-2{\mathrm{{i}}}\beta } \bigl(q(z_{0}) \bigr)^{2} + \alpha {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } z_{0}q'(z_{0}) \bigr] \\ &\quad = {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } (\gamma + {\mathrm{{i}}}\rho )^{2} + ( \alpha _{1} + {\mathrm{{i}}}\alpha _{2} ) m\sigma \\ &\quad = \bigl(\gamma ^{2}-\rho ^{2} \bigr) \cos \beta + 2 \gamma \rho \sin \beta + m \sigma \alpha _{1} \\ &\qquad{} + {\mathrm{{i}}} \bigl[ 2\gamma \rho \cos \beta - \bigl( \gamma ^{2}-\rho ^{2} \bigr)\sin \beta +m \sigma \alpha _{2} \bigr]. \end{aligned} $$

Hence taking real parts in the above, and from \(\sigma \alpha _{1} \leq 0\) and \(m \geq 1\), we have

$$ \begin{aligned}[b] \operatorname{Re} \bigl\{ { \mathrm{{e}}}^{{\mathrm{{i}}}\beta } \bigl[ \bigl(p(z_{0}) \bigr)^{2} + \alpha z_{0} p'(z_{0}) \bigr] \bigr\} &= \bigl(\gamma ^{2}-\rho ^{2} \bigr) \cos \beta + 2\gamma \rho \sin \beta + m \sigma \alpha _{1} \\ &\leq \bigl(\gamma ^{2}-\rho ^{2} \bigr) \cos \beta + 2 \gamma \rho \sin \beta + \sigma \alpha _{1}. \end{aligned} $$
(2.22)

Now equation (2.6) gives

$$ \bigl(\gamma ^{2}-\rho ^{2} \bigr) \cos \beta + 2\gamma \rho \sin \beta + \sigma \alpha _{1} = - a_{2} \rho ^{2} + a_{1} \rho + a_{0}, $$
(2.23)

where

$$ a_{2} = \cos \beta + \frac{\alpha _{1}}{2\mu }, \qquad a_{1} = \biggl( 2 \gamma + \frac{ \alpha _{1} }{\mu } \biggr) \sin \beta $$

and

$$ a_{0} = \gamma ^{2} \cos \beta + \frac{ \alpha _{1}(2\gamma \cos \beta -1-\gamma ^{2}) }{ 2\mu }, $$

with \(\mu = \cos \beta -\gamma \).

Clearly, \(a_{2}>0\). Thus we have

$$ -a_{2} \rho ^{2} + a_{1} \rho +a_{0} \leq \frac{ a_{1}^{2} }{ 4a_{2} } + a_{0}, \quad \rho \in \mathbb{R}. $$
(2.24)

Consequently, by (2.22), (2.23), and (2.24), we obtain

$$ \operatorname{Re} \bigl\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \bigl[ \bigl(p(z_{0}) \bigr)^{2} + \alpha z_{0} p'(z_{0}) \bigr] \bigr\} \leq \frac{ a_{1}^{2} }{ 4a_{2} } + a_{0} = \Psi (\alpha ,\beta ,\gamma ). $$

This contradicts (2.21). Therefore we obtain \(q \prec h\) in \(\mathbb{D}\), and it follows that the inequality \(\operatorname{Re}\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta }p(z) \} > \gamma \) holds for all \(z\in \mathbb{D}\). □

Since the condition

$$ \bigl\vert \arg ( w-a\sec \beta ) \bigr\vert < \frac{\pi }{2}-\beta $$

implies

$$ \operatorname{Re} \bigl\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } w \bigr\} > a \quad \text{and}\quad \operatorname{Re} \bigl\{ {\mathrm{{e}}}^{-{\mathrm{{i}}}\beta } w \bigr\} > a, $$

for \(w\in \mathbb{C}\), \(a \in \mathbb{R}\) and \(\beta \in (-\pi /2,\pi /2)\), by noting that \(\Psi (\alpha ,\beta ,\gamma ) = \Psi (\alpha ,-\beta ,\gamma )\), the following result can be obtained from Theorem 2.3.

Theorem 2.4

Let \(\alpha \in \mathbb{C}\) with \(\operatorname{Re}(\alpha )\geq 0\). Assume that \(\Psi (\alpha ,\beta ,\gamma ) < \cos \beta \), where Ψ is given by (2.20). If \(p \in {\mathcal{H}}_{1}\) satisfies

$$ \bigl\vert \arg \bigl\{ \bigl(p(z) \bigr)^{2} + \alpha zp'(z) - \Psi (\alpha , \beta ,\gamma ) \sec \beta \bigr\} \bigr\vert < \frac{\pi }{2} -\beta , \quad z\in \mathbb{D}, $$

then

$$ \bigl\vert \arg \bigl(p(z)-\gamma \bigr) \bigr\vert < \frac{\pi }{2} -\beta , \quad z \in \mathbb{D}. $$

Taking \(\alpha =1\) and \(p(z)=zf'(z)/f(z)\), \(f \in {\mathcal{A}}\), in Theorems 2.3 and 2.4 we have the following corollary.

Corollary 2.3

Assume that \(\Psi (1,\beta ,\gamma ) < \cos \beta \), where Ψ is given by (2.20). If \(f \in {\mathcal{A}}\) satisfies

$$ \operatorname{Re} \biggl\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \biggl( \frac{zf'(z)}{f(z)} \biggr) \biggl( 1+ \frac{zf''(z)}{f'(z)} \biggr) \biggr\} > \Psi (1,\beta ,\gamma ), \quad z\in \mathbb{D}, $$

then f is a β-spirallike function of order γ in \(\mathbb{D}\). If \(f \in {\mathcal{A}}\) satisfies

$$ \biggl\vert \arg \biggl\{ \biggl( \frac{zf'(z)}{f(z)} \biggr) \biggl( 1+ \frac{zf''(z)}{f'(z)} \biggr) - \Psi (1,\beta ,\gamma ) \sec \beta \biggr\} \biggr\vert < \frac{\pi }{2}-\beta , \quad z\in \mathbb{D}, $$

then f is strongly starlike of order \(1-(2/\pi )\beta \) and type γ in \(\mathbb{D}\).

Example 2.3

Let \(a\in \mathbb{C}\), and define a function \(f_{a}:\mathbb{D}\rightarrow \mathbb{C}\) by \(f_{a}(z)=z/(1-az)\). Then, since \(|z|<1\), we have

$$ \begin{aligned} &\operatorname{Re} \biggl\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \biggl( \frac{zf'(z)}{f(z)} \biggr) \biggl( 1+ \frac{zf''(z)}{f'(z)} \biggr) - \cos \beta \biggr\} \\ &\quad = \operatorname{Re} \biggl\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \frac{ az(3-az) }{ (1-az)^{2} } \biggr\} \geq - \frac{ \vert a \vert \vert z \vert \vert 3-az \vert }{ \vert 1-az \vert ^{2} } > - \frac{ \vert a \vert (3+ \vert a \vert ) }{ (1- \vert a \vert )^{2} }, \end{aligned} $$

or, equivalently,

$$ \operatorname{Re} \biggl\{ {\mathrm{{e}}}^{{\mathrm{{i}}}\beta } \biggl( \frac{zf'(z)}{f(z)} \biggr) \biggl( 1+ \frac{zf''(z)}{f'(z)} \biggr) \biggr\} > \cos \beta - \frac{ \vert a \vert (3+ \vert a \vert ) }{ (1- \vert a \vert )^{2} }. $$

By Corollary 2.3, \(f_{a}\) is a β-spirallike function of order γ provided

$$ \frac{ \vert a \vert (3+ \vert a \vert ) }{ (1- \vert a \vert )^{2} } \leq \cos \beta - \Psi (1,\beta , \gamma ), $$

where Ψ is given by (2.20). In particular, if

$$ \vert a \vert \leq \frac{1}{76}(-239+3\sqrt{6769}) =: \tau = 0.1029 \cdots , $$

then \(f_{a}\) is \((\pi /3)\)-spirallike function of order \(1/3\). Indeed, when \(\beta =\pi /3\) and \(\gamma =1/3\), we have \(\cos \beta - \Psi (1,\beta ,\gamma ) = 25/63\). Solving the inequality \(|a|(3+|a|)/(1-|a|)^{2} \leq 25/63\) gives us to get \(|a| \leq \tau \).

Example 2.4

Let \(a\in \mathbb{C}\) be given, and let \(g_{a}(z) = z/(1-az)^{2}\), \(z\in \mathbb{D}\). Then, from a similar computation with Example 2.3 and Corollary 2.3, we have that \(g_{a}\) is a β-spirallike function of order γ, if

$$ \frac{1+4 \vert a \vert + \vert a \vert ^{2}}{(1- \vert a \vert )^{2}} \leq \cos \beta - \Psi (1,\beta , \gamma ). $$

Concluding remarks and observations

In the present investigation, we have found several conditions for Carathéodory functions by using a technique of the first-order differential subordination. In particular, one can obtain conditions for Carathéodory functions of order γ (\(0 < \gamma \leq 1\)) and for tilted Carathéodory functions by angle β (\(-\pi /2 < \beta < \pi /2\)). We have applied these results to obtain new criteria for geometric properties such as spirallikeness and strongly starlikeness, and several examples were given here.

We conclude this paper by remarking that the results here reduce the earlier conditions [13] for Carathéodory functions. Also, as the examples in this paper show, the first-order differential subordination with the conformal mapping \(\varphi _{\beta ,\gamma }\) defined by (2.1) gives some nice criteria for spirallike functions and strongly starlike functions. This observation will indeed apply to any attempt to produce the conditions for other geometric properties such as convexity, q-starlikeness, etc. [13, 10, 22, 29, 30].

Availability of data and materials

There is no additional data required for finding the results of this paper.

References

  1. Agarwal, P., Agarwal, P., Ruzhansky, M.: Special Functions and Analysis of Differential Equations. Chapman & Hall, London (2020)

    Book  Google Scholar 

  2. Araci, S., Acikgoz, M.: Computation of Nevanlinna characteristic functions derived from generating functions of some special numbers. Journal of inequalities and applications. 2018(1), 1

  3. Attiya, A.A., Lashin, A.M., Ali, E.E., Agarwal, P.: Coefficient bounds for certain classes of analytic functions associated with faber polynomial. Symmetry 13(2), 302 (2021)

    Article  Google Scholar 

  4. Brannan, D.A., Kirwan, W.E.: On some classes of bounded univalent functions. J. Lond. Math. Soc. 2–1(1), 431–443 (1969)

    MathSciNet  Article  Google Scholar 

  5. Darus, M., Hussain, S., Raza, M., Sokół, J.: On a subclass of starlike functions. Results Math. 73(1), Article ID 22 (2018). https://doi.org/10.1007/s00025-018-0771-3

    MathSciNet  Article  MATH  Google Scholar 

  6. Ebadian, A., Sokół, J.: On the subordination and superordination of strongly starlike functions. Math. Slovaca 66(4), 815–822 (2016). https://doi.org/10.1515/ms-2015-0184

    MathSciNet  Article  MATH  Google Scholar 

  7. Frasin, B.A.: Sufficient conditions for λ-spirallike and λ-Robertson functions of complex order. J. Math. 2013, Article ID 194053 (2013). https://doi.org/10.1155/2013/194053

    MathSciNet  Article  MATH  Google Scholar 

  8. Goodman, A.W.: Univalent Functions. Mariner, Tampa (1983)

    MATH  Google Scholar 

  9. Hotta, I., Nunokawa, M.: On strongly starlike and convex functions of order α and type β. Mathematica 53(76), 51–56 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Kashuri, A., Araci, S.: Integral inequalities for the strongly-generalized nonconvex function. Appl. Math. Inf. Sci. 14(2), 243–248 (2020)

    MathSciNet  Article  Google Scholar 

  11. Kim, I.H., Cho, N.E.: Sufficient conditions for Carathéodory functions. Comput. Math. Appl. 59, 2067–2073 (2010)

    MathSciNet  Article  Google Scholar 

  12. Kim, Y.C., Sugawa, T.: The Alexander transform of a spirallike function. J. Math. Anal. Appl. 325(1), 608–611 (2007). https://doi.org/10.1016/j.jmaa.2006.01.077

    MathSciNet  Article  MATH  Google Scholar 

  13. Kwon, O.S., Sim, Y.J.: Sufficient conditions for Carathéodory functions and applications to univalent functions. Math. Slovaca 69(5), 1065–1076 (2019). https://doi.org/10.1515/ms-2017-0290

    MathSciNet  Article  MATH  Google Scholar 

  14. Miller, S.S., Mocanu, P.T.: Marx-Strohhächker differential subordination systems. Proc. Am. Math. Soc. 99, 527–534 (1987)

    MATH  Google Scholar 

  15. Miller, S.S., Mocanu, P.T.: Differential Subordination: Theory and Applications. Series on Monographs and Textbooks in Pure and Applied Mathematics, vol. 225. Dekker, New York (2000)

    Book  Google Scholar 

  16. Montel, P.: Leçons sur les fonctions univalentes on multivalentes. Gauthier-Villars, Paris (1933)

    MATH  Google Scholar 

  17. Nunokawa, M.: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad. Ser. A 69, 234–237 (1993)

    MathSciNet  Article  Google Scholar 

  18. Nunokawa, M., Kwon, O.S., Sim, Y.J., Cho, N.E.: Sufficient conditions for Carathéodory functions. Filomat 32(3), 1097–1106 (2018)

    MathSciNet  Article  Google Scholar 

  19. Nunokawa, M., Sokół, J.: New conditions for starlikeness and strongly starlikeness of order alpha. Houst. J. Math. 43(2), 333–344 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Nunokawa, M., Sokół, J.: Some applications of first-order differential subordinations. Math. Slovaca 67(4), 939–944 (2017). https://doi.org/10.1515/ms-2017-0022

    MathSciNet  Article  MATH  Google Scholar 

  21. Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, Gottingen (1975)

    MATH  Google Scholar 

  22. Ruzhansky, M., Cho, Y.J., Agarwal, P., Area, I.: Advances in Real and Complex Analysis with Application. Springer, Singapore (2017)

    Book  Google Scholar 

  23. Sim, Y.J., Kwon, O.S., Cho, N.E., Srivastava, H.M.: Some sets of sufficient conditions for Carathéodory functions. J. Comput. Anal. Appl. 21(7), 1243–1254 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Špaček, L.: Contribution à la theorie des fonctions univalentes. Čas. Pěst. Mat. 62, 12–19 (1932)

    Article  Google Scholar 

  25. Stankiewicz, J.: Quelques problémes extrémaux dans les classes des fonctions α-angulairement étoilées. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 20, 59–75 (1966)

    MathSciNet  MATH  Google Scholar 

  26. Tuneski, N.: Some simple sufficient conditions for starlikeness and convexity. Appl. Math. Lett. 22(5), 693–697 (2009). https://doi.org/10.1016/j.aml.2008.08.006

    MathSciNet  Article  MATH  Google Scholar 

  27. Wang, L.M.: The tilted Carathéodory class and its applications. J. Korean Math. Soc. 49(4), 671–686 (2012). https://doi.org/10.4134/JKMS.2012.49.4.671

    MathSciNet  Article  MATH  Google Scholar 

  28. Xu, Q., Lu, S.: The Alexander transformation of a subclass of spirallike functions of type β. J. Inequal. Pure Appl. Math. 10(1), Article ID 17 (2009)

    MathSciNet  MATH  Google Scholar 

  29. Yassen, M.F., Attiya, A.A., Agarwal, P.: Subordination and superordination properties for certain family of analytic functions associated with Mittag-Leffler function. Symmetry 12(10), 1724 (2020)

    Article  Google Scholar 

  30. Zhou, S.S., Areshi, M., Agarwal, P., Shah, N.A., Chung, J.D., Nonlaopon, K.: Analytical analysis of fractional-order multi-dimensional dispersive partial differential equations. Symmetry 13(6), 939 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861).

Funding

There is no funding available for the publication of this paper.

Author information

Authors and Affiliations

Authors

Contributions

All authors have equal contribution in this article. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Young Jae Sim.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cho, N.E., Kwon, O.S. & Sim, Y.J. Differential inequalities for spirallike and strongly starlike functions. Adv Differ Equ 2021, 511 (2021). https://doi.org/10.1186/s13662-021-03670-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-021-03670-9

MSC

  • 30C45
  • 30C80

Keywords

  • Carathéodory functions
  • Differential subordination
  • Starlike functions
  • Spirallike functions
  • Strongly starlike functions