In this section we introduce and examine the notion of starlikeness and convexity related to the domains bounded by the hyperbolas \(H(s)\). We indicate how the properties of the domains \(\mathbb{H}(s)\) and class \(\mathcal{P}(\mathfrak {q}_{s})\) influence on the properties of a newly introduced classes.
Definition 3.1
Let \(\mathcal{ST}_{\mathrm{hpl}}(s)\) denote the subfamily of \(\mathcal{S}\) consisting of the functions f satisfying the condition
$$ \frac{zf'(z)}{f(z)}\prec \mathfrak {q}_{s}(z)\quad (z\in \mathbb{D} ), $$
(3.1)
and let \(\mathcal{CV}_{\mathrm{hpl}}(s)\) be a class of analytic functions f such that
$$ 1+\frac{zf''(z)}{f'(z)}\prec \mathfrak {q}_{s}(z) \quad (z\in \mathbb{D} ). $$
(3.2)
Geometrically, the condition (3.1) and (3.2) means that the expression \(zf'(z)/f(z)\) (or \(1+zf''(z)/f'(z)\), resp.) lies in a domain \(\mathbb{H}(s)\). Since \(\mathbb{H}(s)=\mathfrak {q}_{s}(\mathbb{D})\) is contained in a right halfplane, we deduce that \(\mathcal{ST}_{\mathrm{hpl}}(s)\) is a proper subset of a class of a starlike functions \(\mathcal{ST}\) (and \(\mathcal{CV}_{\mathrm{hpl}}(s)\subset \mathcal{CV}\), resp.). Furthermore, the properties of \(\mathbb{H}(s)\) considered in Sect. 1, yield
$$\begin{aligned}& \mathcal{ST}_{\mathrm{hpl}}(s)\subset \mathcal{ST}(\alpha ) \quad \text{and}\quad \mathcal{CV}_{\mathrm{hpl}}(s)\subset \mathcal{CV}(\alpha ) \quad \text{for } 0 \le \alpha \leq 2^{-s}, 0< s\le 1, \\& \mathcal{ST}_{\mathrm{hpl}}(s)\subset \mathcal{ST}_{\gamma }\quad \text{and} \quad \mathcal{CV}_{\mathrm{hpl}}(s)\subset \mathcal{CV}_{\gamma } \quad \text{for } 0< s \le \gamma \le 1. \end{aligned}$$
The above inclusions additionally give
$$ \mathcal{ST}_{\mathrm{hpl}}(s) \subset \mathcal{ST}_{s} \cap \mathcal{ST} \bigl(1/2^{s} \bigr),\qquad \mathcal{CV}_{\mathrm{hpl}}(s) \subset \mathcal{CV}_{s} \cap \mathcal{CV} \bigl(1/2^{s} \bigr),\quad 0< s\le 1. $$
Also, geometric properties of \(\varOmega _{k}\) and \(\mathbb{H}(s)\) given by (1.1), (1.2), (1.3) and also (1.4) imply
$$\begin{aligned}& \mathcal{ST}_{\mathrm{hpl}}(s)\subset k\mbox{-}\mathcal{ST}\quad \text{and}\quad \mathcal{CV}_{\mathrm{hpl}}(s)\subset k\mbox{-}\mathcal{CV} \quad \text{for } 0 \le k \le 2^{-s}, 0< s\le 1, \\& k\mbox{-}\mathcal{ST}\subset \mathcal{ST}_{\mathrm{hpl}}(s) \quad \text{and}\quad k \mbox{-}\mathcal{CV}\subset \mathcal{CV}_{\mathrm{hpl}}(s) \quad \text{for } k \ge 2^{s}-1, 0< s\le 1, \\& k\mbox{-}\mathcal{ST}\cap \mathcal{ST}_{\mathrm{hpl}}(s) \neq \emptyset \quad \text{and}\quad k\mbox{-}\mathcal{CV}\cap \mathcal{CV}_{\mathrm{hpl}}(s) \neq \emptyset \quad \text{for } 2^{-s}< k < 2^{s}-1, 0< s\le 1. \end{aligned}$$
Moreover
$$ k\mbox{-}\mathcal{ST}\not \subset \mathcal{ST}_{\mathrm{hpl}}(s)\quad \text{and}\quad \mathcal{ST}_{\mathrm{hpl}}(s)\not \subset k\mbox{-} \mathcal{ST} \quad \text{for } 2^{-s}< k < 2^{s}-1, 0< s\le 1, $$
and
$$ k\mbox{-}\mathcal{CV}\not \subset \mathcal{CV}_{\mathrm{hpl}}(s) \quad \text{and}\quad \mathcal{CV}_{\mathrm{hpl}}(s)\not \subset k\mbox{-} \mathcal{CV}\quad \text{for } 2^{-s}< k < 2^{s}-1, 0< s\le 1. $$
Since \(\mathfrak {q}_{s}\) is the extremal function in \(\mathcal{P}(\mathfrak {q}_{s})\), the obvious integral representation of \(\mathcal{ST}_{\mathrm{hpl}}(s)\) and \(\mathcal{CV} _{\mathrm{hpl}}(s)\) immediately follows.
Theorem 3.2
A function
f
is in the class
\(\mathcal{ST}_{\mathrm{hpl}}(s)\)
if, and only if there exists
p
such that
\(p\prec \mathfrak {q}_{s}\), and
$$ f(z)=z\exp \biggl( \int _{0}^{z} \frac{p(t)-1}{t} \,\mathrm{d}t \biggr). $$
Theorem 3.3
A function
f
is in the class
\(\mathcal{CV}_{\mathrm{hpl}}(s)\)
if, and only if there exists
p
such that
\(p\prec \mathfrak {q}_{s}\), and
$$ f(z)= \int _{0}^{z} \exp \biggl( \int _{0}^{x} \frac{p(t)-1}{t} \,\mathrm{d}t \biggr) \,\mathrm{d}x. $$
Suppose that \(\varPhi _{s,n}\in \mathcal{ST}_{\mathrm{hpl}}(s)\) is such that
$$ \frac{z\varPhi '_{s,n}(z)}{\varPhi _{s,n}(z)} = \frac{1}{ (1-z ^{n} )^{s}}\quad (z \in \mathbb{D}, n=1, 2, \ldots ). $$
(3.3)
Then the functions \(\varPhi _{s,n}(z)\) are of the form
$$\begin{aligned} \varPhi _{s,n}(z) =&z\exp \biggl( \int _{0}^{z} \frac{ \mathfrak{q}_{s}(t^{n})-1}{t} \,\mathrm{d}t \biggr)=z+ \frac{s}{n}z^{n+1}+ \frac{(n+2)s^{2}+ns}{4n^{2}} z^{2n+1} \\ &{} + \frac{4n^{2}s+(9n+6n^{2})s^{2}+(2n^{2}+9n+6)s^{3}}{36n ^{3}} z^{3n+1}\cdots , \end{aligned}$$
(3.4)
and these are extremal functions for different problems in the class \(\mathcal{ST}_{\mathrm{hpl}}(s)\). For instance
$$\begin{aligned} \varPhi _{s}(z) =& \varPhi _{s,1}(z)=z \exp \biggl( \int _{0}^{z} \frac{\mathfrak {q}_{s}(t)-1}{t} \,\mathrm{d}t \biggr) \\ =&z+s z^{2}+ \frac{3s^{2}+s}{4} z^{3}+\frac{17s^{3}+15s ^{2}+4s}{36} z^{4} \cdots \quad (z\in \mathbb{D} ). \end{aligned}$$
(3.5)
The special case \(s=1/2\) gives
$$\begin{aligned} \varPhi _{1/2,n}(z) &=z\exp \biggl( \int _{0}^{z} \frac{ \mathfrak{q}_{1/2}(t^{n})-1}{t} \,\mathrm{d}t \biggr)=z \biggl(\frac{4}{ (1+\sqrt{1-z^{n}} )^{2}} \biggr)^{1/n}, \\ &=z+\frac{1}{2n}z^{n+1}+ \frac{7-2n}{16n^{2}} z^{2n+1}+\cdots \end{aligned}$$
and
$$\begin{aligned} \varPhi _{1/2}(z)&=\varPhi _{1/2,1}(z) =z\exp \biggl( \int _{0}^{z} \frac{ \mathfrak{q}_{1/2}(t)-1}{t} \,\mathrm{d}t \biggr)=\frac{4z}{ (1+\sqrt{1-z} )^{2}} \\ &=z+\frac{1}{2} z^{2}+\frac{5}{16}z^{3}+ \cdots \quad (z\in \mathbb{D} ). \end{aligned}$$
Also, suppose that \(K_{s,n}\in \mathcal{CV}_{\mathrm{hpl}}(s)\) is such that
$$ 1+\frac{zK''_{s,n}(z)}{K'_{s,n}(z)} = \frac{1}{ (1-z ^{n} )^{s}}\quad (z \in \mathbb{D}, n=1, 2, \ldots ). $$
(3.6)
Then the functions \(K_{s,n}(z)\) are of the form
$$\begin{aligned} K_{s,n}(z) &= \int _{0}^{z} \exp \biggl( \int _{0}^{w} \frac{\mathfrak{q}_{s}(t^{n})-1}{t} \,\mathrm{d}t \biggr) \,\mathrm{d}w \\ &= z+\frac{s}{n(n+1)}z^{n+1} + \frac{(n+2)s^{2}+ns}{4n ^{2}(2n+1)} z^{2n+1} \\ &\quad {} + \frac{4n^{2}s+(9n+6n^{2})s^{2}+(2n^{2}+9n+6)s ^{3}}{36n^{3}(3n+1)} z^{3n+1}\cdots , \end{aligned}$$
and the \(K_{s,n}\) are extremal functions for various problems in the class \(\mathcal{CV}_{\mathrm{hpl}}(s)\). The most interesting special cases are for \(n=1\)
$$\begin{aligned} K_{s}(z) &=K_{s,1}(z)= \int _{0}^{z} \exp \biggl( \int _{0}^{w} \frac{\mathfrak{q}_{s}(t)-1}{t} \,\mathrm{d}t \biggr) \,\mathrm{d}w= \int _{0}^{z} \frac{ \varPhi _{s}(w)}{w} \,\mathrm{d}w \\ &=z+\frac{s}{2} z^{2}+ \frac{3s^{2}+s}{12} z^{3}+ \frac{17s^{3}+15s ^{2}+4s}{144} z^{4} \cdots \quad (z\in \mathbb{D} ) \end{aligned}$$
(3.7)
and for \(s=1/2\), \(n=1\)
$$\begin{aligned} K_{1/2}(z)=K_{1/2,1}(z) &= \int _{0}^{z}\frac{\varPhi _{1/2}(w)}{w} \,\mathrm{d} w \\ &=8\log \biggl(\frac{2}{1+\sqrt{1-z}} \biggr)+4 \biggl(\frac{\sqrt{1-z}-1}{ \sqrt{1-z}+1} \biggr) \\ &=z+\frac{1}{4} z^{2}+\frac{5}{48}z^{3}+ \frac{7}{128}z^{4}+ \frac{21}{640}z^{5}+ \cdots (z\in \mathbb{D} ). \end{aligned}$$
From Theorems 3.2 and 3.3 we obtain the following corollary.
Corollary 3.4
If
\(f\in \mathcal{CV}(1-s/2)\), then
\(f'\prec \mathfrak {q}_{s}\)
in
\(\mathbb{D}\). Thus
f
is univalent in
\(\mathbb{D}\). The functions
H
and
G
defined by
$$ H(z)=z\exp \biggl( \int _{0}^{z}\frac{f'(t)-1}{t} \,\mathrm{d}t \biggr),\qquad G(z)= \int _{0}^{z} \exp \biggl( \int _{0}^{x} \frac{f'(t)-1}{t} \,\mathrm{d}t \biggr) \,\mathrm{d}x, $$
belong to
\(\mathcal{ST}_{\mathrm{hpl}}(s)\)
and
\(\mathcal{CV}_{\mathrm{hpl}}(s)\), respectively.
Examples
Defined classes \(\mathcal{ST}_{\mathrm{hpl}}(s)\) and \(\mathcal{CV}_{\mathrm{hpl}}(s)\) are nonempty. The integral representation given in Theorems 3.2 and 3.3 provides various examples of functions of those classes. For example, if \(q_{1}(z) = 1+B_{1}z\) with \(0< B_{1}<1-2^{-s}\), then \(q_{1}\prec \mathfrak {q}_{s}\). Hence \(f_{1}(z) = z\exp (B_{1}z) \in \mathcal{ST}_{\mathrm{hpl}}(s)\), and \(g_{1}(z) = (e^{B_{1}z}-1)/B_{1} \in \mathcal{CV}_{\mathrm{hpl}}(s)\) (for \(0< B_{1}<1-2^{-s}\)), respectively. Since
$$ f_{2}(z)=\frac{z}{1-B_{2}z},\qquad f_{3}(z)= \frac{z}{ (1-B _{3}z )^{2}}, $$
for \(0<|B_{2}|\le s/(2-s)\), \(0<|B_{3}|\le s/(4-s)\) are starlike of order \((1-s/2)\), applying Corollary 3.4 we see that the appropriate functions
$$ H_{2}(z)= \frac{z}{1-B_{2}z}, \qquad H_{3}(z)= \frac{z}{1-B_{3}z}\exp \biggl(\frac{B_{3}z}{1-B_{3}z} \biggr) $$
belong to the class \(\mathcal{ST}_{\mathrm{hpl}}(s)\), and the corresponding functions will be elements of \(\mathcal{CV}_{\mathrm{hpl}}(s)\).
Theorem 3.5
The function
\(\varPhi _{s,n}(z)\), defined by (3.4), is normalized, univalent and convex in one direction in
\(\mathbb{D}\)
for
\(n=1,2, \ldots , \lfloor {(2^{1-s}+1)/s\rfloor }\), and convex for
\(n=1,2, \ldots , \lfloor {2^{1-s}/s}\rfloor \), where
\(\lfloor {\cdot }\rfloor \)
is the floor function.
Proof
It is a simple matter to check that \(\varPhi _{s,n}(z)\) is normalized by \(\varPhi _{s,n}(0)=\varPhi '_{s,n}(0)-1=0\). By a definition (3.3) of \(\varPhi _{s,n}(z)\), we have
$$ \frac{\varPhi '_{s,n}(z)}{\varPhi _{s,n}(z)}=\frac{1}{z (1-z ^{n} )^{s}}. $$
Hence
$$ \Re \biggl(1+\frac{z\varPhi ''_{s,n}(z)}{\varPhi '_{s,n}(z)} \biggr)=\Re \biggl(ns\frac{z^{n}}{1-z^{n}}+ \frac{1}{(1-z^{n})^{s}} \biggr), $$
and from the above we see that
$$ \Re \biggl(1+\frac{z\varPhi ''_{s,n}(z)}{\varPhi '_{s,n}(z)} \biggr)> 2^{-s}-\frac{ns}{2}. $$
In order to get univalence, it is convenient to use the result by Umezawa [12], with the requirement \(\Re (1+zh''(z)/h'(z) ) > -1/2\) for univalence of h in \(\mathbb{D}\). That condition holds if
$$ 2^{-s}-\frac{ns}{2}\ge -\frac{1}{2}, $$
which is satisfied for \(n=1,2, \ldots , \lfloor {(2^{1-s}+1)/s \rfloor }\). We recall that the same condition gives the convexity in one direction. Furthermore, \(\varPhi _{s,n}(z)\) is convex in \(\mathbb{D} \) if \(\Re (1+z\varPhi _{s,n}''(z)/\varPhi _{s,n}'(z) ) >0\) holds for \(n\le 2^{1-s}/s\), i.e. for \(n=1,2, \ldots , \lfloor {2^{1-s}/s} \rfloor \). □
We have the following from the results in [7], and Lemma 2.1.
Corollary 3.6
If
\(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\)
and
\(|z|=r<1\), then
$$\begin{aligned}& -\varPhi _{s}(-r)\leq \bigl\vert f(z) \bigr\vert \leq \varPhi _{s}(r), \\& \varPhi '_{s}(-r)\leq \bigl\vert f'(z) \bigr\vert \leq \varPhi '_{s}(r), \\& \bigl\vert \operatorname{Arg}\bigl\{ {f(z)}/{z} \bigr\} \bigr\vert \leq \max _{ \vert z \vert =r}\operatorname{Arg}\bigl\{ {\varPhi _{s}(z)}/{z} \bigr\} , \\& {f(z)}/{z}\prec {\varPhi _{s}(z)}/{z}\quad (z\in \mathbb{D} ). \end{aligned}$$
Equality holds for some
\(z_{0}\neq 0\)
if and only if
f
is a rotation of
\(\varPhi _{s}\), where
\(\varPhi _{s}\)
is given by (3.5).
Also, if
\(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\), then either
f
is a rotation of
\(\varPhi _{s}\)
or
$$ \bigl\{ w\in \mathbb{C} : \vert w \vert \leq -\varPhi _{s}(-1) \bigr\} \subset f(\mathbb{D}). $$
Here
\(-\varPhi _{s}(-1)\)
is understood to be the limit of
\(-\varPhi _{s}(-r)\)
as
r
tends to 1.
Similar results hold for functions of \(\mathcal{CV}_{\mathrm{hpl}}(s)\).
Corollary 3.7
If
\(f\in \mathcal{CV}_{\mathrm{hpl}}(s)\)
and
\(|z|=r<1\), then
$$\begin{aligned}& -K_{s}(-r)\leq \bigl\vert f(z) \bigr\vert \leq K_{s}(r), \\& K'_{s}(-r)\leq \bigl\vert f'(z) \bigr\vert \leq K'_{s}(r), \\& \bigl\vert \operatorname{Arg}\bigl\{ f'(z) \bigr\} \bigr\vert \leq \max _{ \vert z \vert =r} \operatorname{Arg}\bigl\{ K'_{s}(z) \bigr\} , \\& f'(z)\prec K'_{s}(z) \quad (z\in \mathbb{D} ). \end{aligned}$$
Equality holds for some
\(z_{0}\neq 0\)
if and only if
f
is a rotation of
\(K_{s}\), where
\(K_{s}\)
is given by (3.7).
Also, if
\(f\in \mathcal{CV}_{\mathrm{hpl}}(s)\), then either
f
is a rotation of
\(K_{s}\)
or
$$ \bigl\{ w\in \mathbb{C} : \vert w \vert \leq -K_{s}(-1) \bigr\} \subset f(\mathbb{D}). $$
Here
\(-K_{s}(-1)\)
is understood to be the limit of
\(-K_{s}(-r)\)
as
r
tends to 1.
Theorem 3.8
Let
\(r_{0}\)
denote the positive root of the equation
$$ \frac{1}{ (1+r )^{s}}= \frac{rs}{1-r}\quad (0\le r< 1 ). $$
(3.8)
If
\(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\), then
f
is convex in the disk
\(|z|< r_{0}\).
Proof
Let \(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\). Then from Definition 3.1 we obtain
$$ \frac{zf'(z)}{f(z)}=\frac{1}{ (1-w(z) )^{s}}\quad (z\in \mathbb{D} ), $$
where \(w\in \mathcal{B}\). Application of Lemma 2.1 and the well-known inequality for Schwarz functions \(|w'(z)|\le (1-|w(z)|^{2})/(1-|z|^{2})\) gives
$$\begin{aligned} \Re \biggl(1+\frac{zf''(z)}{f'(z)} \biggr) &\geq \Re { \frac{1}{ (1-w(z) ) ^{s}}}-s \vert z \vert \frac{1- \vert w(z) \vert ^{2}}{ [1- \vert w(z) \vert ] [1- \vert z \vert ^{2} ]} \\ &\geq \frac{1}{ (1+ \vert z \vert ) ^{s}}-\frac{s \vert z \vert }{1- \vert z \vert }. \end{aligned}$$
The function \(\psi (r)= \frac{1}{ (1+r )^{s}}-\frac{rs}{1-r}\), where \(r=|z| \in [0,1)\) is decreasing in \([0,1)\) with \(\psi (0)=1\) and \(\psi (1^{-})=- \infty \). Thus, there exists a unique \(r_{0}\in (0,1)\) such that \(\psi (r_{0})=0\), and for \(0\le r< r_{0}\) we have \(0< \psi (r)\le 1\). This inequality is equivalent to
$$ \frac{1}{ (1+r )^{s}}> \frac{rs}{1-r}\quad \bigl(r\in [0,1)\bigr), $$
which is satisfied for \(|z|< r_{0}\), where \(r_{0}\) is the only real positive root of (3.8). □
Theorem 3.9
If
\(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\), then there exists
\(\alpha >1\)
such that
$$ \Re \biggl\{ 1+\frac{zf''(z)}{f'(z)} \biggr\} < \alpha \quad \bigl( \vert z \vert =r < 1 \bigr). $$
Proof
Let \(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\). Then, from Definition 3.1, we obtain
$$ \frac{zf'(z)}{f(z)}=\frac{1}{ (1-w(z) )^{s}} \quad (z\in \mathbb{D} ), $$
where \(w\in \mathcal{B}\). By Lemma 2.1 and the inequality for Schwarz functions it follows that
$$\begin{aligned} \Re \biggl(1+\frac{zf''(z)}{f'(z)} \biggr) &\le \Re { \frac{1}{ (1-w(z) ) ^{s}}}+s \vert z \vert \frac{1- \vert w(z) \vert ^{2}}{ [1- \vert w(z) \vert ] [1- \vert z \vert ^{2} ]} \\ &\le \frac{1}{ (1- \vert z \vert ) ^{s}}+\frac{s \vert z \vert }{1- \vert z \vert }. \end{aligned}$$
The function \(g(r)= \frac{1}{ (1-r )^{s}}+\frac{rs}{1-r}\), where \(r=|z| \in (0,1)\), is increasing in \((0,1)\) and \(g(0)=1\). Thus for \(z\in \mathbb{D}\) we have \(g(|z|)>1\). □