Abstract
In this paper we show several sufficient conditions for close-to-convex functions to be strongly starlike of some order. The results continue the line of study from the first author’s paper on the order of strong starlikeness of strongly convex functions, (Nunokawa in Proc Japan Acad Ser A 69(7):234–237, 1993). Also it appears an small improvement of a certain classical results of Ch. Pommerenke. As an application, we also derive estimates for the radii of star-likeness for close-to-convex functions.
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1 Introduction
Let \({\mathcal {H}}\) denote the class of analytic functions in the unit disc \({{\mathbb {D}}}=\{z:\ |z|<1\}\) on the complex plane \({\mathbb {C}}\). Let \({\mathcal {A}}\) denote the subclass of \({\mathcal {H}}\) consisting of functions normalized by \(f(0)=0\), \(f'(0)=1\). The set of all functions \(f\in {\mathcal {A}}\) that are convex univalent in \({\mathbb {D}}\) we denote by \({\mathcal {K}}\). The set of all functions \(f\in {\mathcal {A}}\) that are starlike univalent in \({\mathbb {D}}\) with respect to the origin we denote by \({\mathcal {S}^*}\). Recall that a set \(E\subset {\mathbb {C}}\) is said to be starlike with respect to a point \(w_0\in E\) if and only if the line segment joining \(w_0\) to any other point \(w\in E\) lies entirely in E. A set E is said to be convex if and only if it is starlike with respect to each of its points, that is if and only if the linear segment joining any two points of E lies entirely in E. An univalent function f maps \({\mathbb {D}}\) onto a convex domain E if and only if [11]
Such a function f is said to be convex in \({\mathbb {D}}\) (or briefly convex). In [8] Sakaguchi proved that if \(f\in \mathcal {A}\) and \(g\in \mathcal {S}^*\), then
This result was also generalized, see [2] and [7]. In [6] Pommerenke established a formula for \(\beta =\beta (\alpha )\) such that
where \(f(z)\in \mathcal {K}_{\alpha }\). This is a generalization of the relation of type (1.1) because is in the class \(\mathcal {K}_{\alpha }\)\(\alpha \), \(0<\alpha \le 1\) whenever \(f(z)\in {\mathcal {A}}\) and there exist a function \(g(z)\in \mathcal {K}\) such that
Here we understand that \(\mathrm{Arg} w\) is a number in \((-\pi , \pi ]\). It is known that if \(f(z)\in \mathcal {K}_{\alpha }\), then f(z) is close-to-convex and so f(z) is univalent in \(\mathbb {D}\). The class \(f(z)\in \mathcal {K}_{\alpha }\) is called the class of strongly close-to-convex functions of order \(\alpha \).
This result has found many applications. Condition (1.2) with \(g(z)=f(z)\) becomes
and it says that f(z) is a strongly starlike function of order \(\alpha \), \(0<\alpha \le 1\). The class of strongly starlike functions was introduced in [1, 10], we denote this class here by \(\mathcal {S}^*(\alpha )\). One can consider functions satisfying condition (1.3) with \(0<\alpha < 2\) and in this case we will named such functions also strongly starlike of order \(\alpha \), \(0<\alpha < 2\). It is known that if f(z) is strongly starlike of order \(\alpha >1\), then f(z) need not to be univalent in \(\mathbb {D}\).
We say that \(f(z)\in \mathcal {K}_{\beta }^{\alpha }\) whenever \(f(z)\in {\mathcal {A}}\) and there exists a real \(\alpha \), \(0<\alpha \le 1\) and a function \(g(z)\in \mathcal {S}^*(\beta )\cap \mathcal {K}\), \(0<\beta \le 1\), such that
Lemma 1.1
Let \(f(z)\in \mathcal {K}_{\beta }^{\alpha }\) with \(0<2\alpha +\beta \le 1\), \(0<\alpha \le 1\), \(0<\beta \le 1\). Then we have
or f(z) is strongly starlike function, of order \(\beta +2\alpha \).
Proof
If \(f(z)\in \mathcal {K}_{\beta }^{\alpha }\), then f(z) is univalent in \(\mathbb {D}\) and so
exists for all \(z\in \mathbb {D}\). It is known the following Pommerenke’s result [6, Lemma 1, p. 180]: If f(z) is analytic and g(z) is convex in \(\mathbb {D}\), then
Applying this Lemma with \(z_1=0\) gives
it follows that
\(\square \)
We note that a result of the form related to (1.5) was proved in [5, Theorem 2.1].
Lemma 1.2
Let \(h(z)=1+\sum _{n=1}^{\infty }c_nz^n\) be in \(\mathcal {H}\). If
for some \(\alpha >0\), then
and
Proof
From the hypothesis \(h(z)\ne 0\), so the function
is in the class \(\mathcal {H}\). Furthermore, we have
hence
and \(h^{1/\alpha }(z)\) is contained in the circle with the radius and center
respectively. From this, we obtain (1.7) and (1.8). \(\square \)
If we take \(g(z)=z\) then Pommerenke’s result (1.5) becomes the following corollary.
Corollary 1.3
Let f(z) be in \(\mathcal {A}\). If
for some \(\alpha \), \(0<\alpha \le 1\), then
In the next Corollary we extend \(\alpha \) to \(1<\alpha <2:\)
Corollary 1.4
Let f(z) be in \(\mathcal {A}\). If
for some \(\alpha \), \(1<\alpha \le 2\), then
Proof
For arbitrary \(z\in \mathbb {D}\) and from the hypothesis \(1<\alpha \le 2\), can connect the point \(f'(z)\) and \(f'(0)=1\) by a line segment as \(\overline{f'(z)f'(0)}=\overline{f'(z)1}\). Applying the same method as in the proof of Pommerenke [6, p. 180], we have
and so, applying the property of integral mean, we have
\(\square \)
To prove the main results, we also need the following generalization of the Nunokawa’s Lemma, [3, 4].
Lemma 1.5
[5] Let p(z) be analytic function in \(|z|<1\) of the form
with \(p(z)\ne 0\) in \(|z|<1\). If there exists a point \(z_0\), \(|z_0|<1\), such that
and
for some \(\alpha >0\), then we have
where
and
where
Theorem 1.6
Let f(z) and g(z) be in \(\mathcal {A}\). Suppose that
for some \(\alpha \) and \(\beta \) such that \(0<\alpha \) and \(0<\beta <1\). Assume also
then
Proof
Let us write
then we have
and
If there exists \(z_0\in \mathbb {D}\) such that
and
then from Lemma 1.5
for some \(k\ge 1\) when \(\mathrm{Arg} \{p(z_0)\}=\alpha \pi /2\) or for some \(k\le -1\) when \(\mathrm{Arg} \{p(z_0)\}=-\alpha \pi /2\). For the case \(\mathrm{Arg} \{p(z_0)\}=\alpha \pi /2\), from (1.12), we have
This contradicts hypothesis (1.11). For the case \(\mathrm{Arg} p(z_0)=-\alpha \pi /2\), applying the same method as the above, we obtain a \(k\le -1\) and so
This contradicts hypothesis (1.11) too and so it completes the proof. \(\square \)
Now, we prove another improvement of Corollary 1.3.
Theorem 1.7
Let \(f(z)=z+\sum _{n=2}^{\infty }a_nz^n\) be in \(\mathcal {A}\). If
for some \(\alpha \), \(0<\alpha \), then
Proof
Let us write
Then \(p\in \mathcal {H}\) and it is the form
If there exists a point \(z_0\), \(|z_0|<1\) such that
and
for some \(\alpha \), \(0<\alpha \), then from [4], we have
for some k, where \(k\ge 1\) when \(\mathrm{Arg}\{p(z_0)\}=\alpha \pi /2\) while \(k\le -1\) when \(\mathrm{Arg}\{p(z_0)\}=-\alpha \pi /2\). From the equality \(f(z)=zp(z)\), we have
and so, for the case \(\mathrm{Arg}\{p(z_0)\}=\alpha \pi /2\), we have
This contradicts the hypothesis (1.16) and for the case \(\mathrm{Arg}\{p(z_0)\}=-\alpha \pi /2\), applying the same method as the above, we have
This also contradicts the hypothesis (1.16) and so we obtain that (1.17) holds true. \(\square \)
Remark 1
Theorem 1.7 shows that
in \(\mathbb {D}\), where \(\beta =\alpha +\frac{2}{\pi }\tan ^{-1}\alpha \), for some \(\alpha \), \(0<\alpha \).
Remark 2
In [6], Pommerenke supposed that \(0< \alpha \le 1\) but in Theorem 1.7 we supposed \(0<\alpha \) only.
Theorem 1.8
Let \(f(z)=z+\sum _{n=2}^{\infty }a_nz^n\) be in \(\mathcal {K}_{\alpha }\) for some \(\alpha \), \(0<\alpha \le 1\). Suppose also that
If
then f(z) is starlike. If
then f(z) is starlike in \(|z|<|z_0|\), where \(|z_0|\) is the smallest positive root of the equation
which has the form
Proof
From Theorem 1.7, we have
Therefore, we have
If \(\alpha \) satisfies (1.19), then (1.23) follows that f(z) is a starlike function. If \(\alpha \) satisfies (1.20), then applying Lemma 1.2, we have
for some small |z|. It follows that f(z) is starlike in \(|z|<|z_0|\), where \(|z_0|\) is positive root of the equation (1.21). Simple calculation shows that
so we obtain (1.22). \(\square \)
Theorem 1.9
Let \(f(z)=z+\sum _{n=2}^{\infty }a_nz^n\) be in \(\mathcal {K}_{\alpha }\) for \(\alpha \), \(0<\alpha \le 1\). Then f(z) is starlike in
Proof
From Lemma 1.1 and hypothesis of the Theorem, we have
and hence from Lemma 1.2, we have
Putting
we have
for |z| as in (1.24). \(\square \)
Remark 3
The radius of starlikeness in the class \(\mathcal {K}_{\alpha }\) may be equal or larger than the above value. This is an open question.
We say that \(f(z)\in \mathcal {K}^{\alpha }_{\beta }\) whenever \(f(z)\in {\mathcal {A}}\) and there exist a real \(0<\alpha \le 2\) and \(0<\beta \le 1\) and a function \(g(z)\in \mathcal {K}\) such that
and
For \(\beta \in (0,1]\) the function
is in \(\mathcal {K}\) because
and
Therefore, a function f(z) such that \(f(z)\equiv g(z)\) is in the class \(\mathcal {K}^{\alpha }_{\beta }\) for all \(0<\alpha \le 2\). For this class \(\mathcal {K}^{\alpha }_{\beta }\) holds the following theorem under some restriction on \(\alpha \) and \(\beta \).
Theorem 1.10
Let \(f(z)=z+\sum _{n=2}^{\infty }a_nz^n\) be in \(\mathcal {K}^{\alpha }_{\beta }\) for some \(\alpha >0\) and \(0<\beta \le 1\). If \(2\alpha +\beta \le 1\), then f(z) is starlike in \(\mathbb {D}\), if \(2\alpha +\beta >1\), then f(z) is starlike in
Proof
From (1.25), (1.26) and from (1.5), we have
If \(2\alpha +\beta \le 1\), then f(z) is starlike in \(\mathbb {D}\) because in this case (1.28) is lees than \(\pi /2\). For the case \(2\alpha +\beta >1\) we can apply the same method as in the proof of Theorem 1.9 to obtain (1.27). \(\square \)
Theorem 1.11
Let f(z), g(z) be in \(\mathcal {A}\). Suppose that
and
where \(0<\alpha -\beta -\gamma \), \(0<\beta \) and \(0<\gamma \). Then we have
Proof
Putting \( p(z)=f(z)/g(z)\), \(z\in \mathbb {D}\), gives \(p(0)=1\) and \(f(z)=p(z)g(z)\). It follows that
If there exists a point \(z_0\in \mathbb {D}\) such that
and
then we have
This contradicts the hypothesis, and it completes the proof. \(\square \)
Some related condition for starlikeness can be found from [9].
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The authors express their sincerest thanks to the referee for the invaluable suggestions.
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Nunokawa, M., Sokół, J. On the order of strong starlikeness and the radii of starlikeness for of some close-to-convex functions. Anal.Math.Phys. 9, 2367–2378 (2019). https://doi.org/10.1007/s13324-019-00340-8
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DOI: https://doi.org/10.1007/s13324-019-00340-8