Abstract
Let \({{\mathcal {A}}}\) denote the family of all functions that are analytic in the unit disk \({\mathbb {D}} := \{ z:\, |z| < 1 \}\) and satisfy \(f(0)=0= f'(0)-1\). Let \({\mathcal {U}} \) denote the subset of functions \(f\in {{\mathcal {A}}}\) which satisfy
and let \({\mathcal {P}}(2)\) be the subclass of all functions \(f\in {\mathcal {A}}\) such that \(f(z)\ne 0\) for \(0<|z|<1\) and
In this paper, a conjecture on the class \({\mathcal {U}} \) and \({\mathcal {P}}(2)\) has been resolved. Furthermore, two sufficient conditions for functions to be univalent are presented.
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1 Introduction
Let \({{\mathcal {A}}}\) denote the family of all functions that are analytic in the unit disk \({{\mathbb {D}}}:= \{ z:\, |z| < 1 \}\) and satisfy \(f(0)=0= f'(0)-1\). Let \({{\mathcal {B}}}\) denote the set of functions \(\omega \) that are analytic in \({{\mathbb {D}}}\) and satisfy \(|\omega (z)|\le 1 (|z|<1)\). Let S be the set of all functions \(f\in {\mathcal {A}}\) that are univalent in \({\mathbb {D}}\). Let \(S^*\) denote the subset of S consisting of all starlike functions. Let \({\mathcal {U}} \) denote the set of all \(f\in {{\mathcal {A}}}\) satisfying the condition
and let \({\mathcal {P}}(2)\) be the subclass of all functions \(f\in {{\mathcal {A}}}\) such that \(f(z)\ne 0\) for \(0<|z|<1\) and
It is known that \({\mathcal {U}}\subset S \)(see [1]). In recent years, the class \({\mathcal {U}} \) were studied in detail (see [2,3,4,5,6]). Obradović and Ponnusamy[3] proved that
For the function f defined by \( \frac{z}{f(z)}=1+\frac{1}{2}z^{3}\) , which belongs to the class \({\mathcal {U}}\), we have that
i.e., \({\mathcal {P}}(2)\)-radius for the above function f is equal to \(\frac{2}{3}\). The authors considered a subclass of the class \({\mathcal {U}}\) and showed that \({\mathcal {P}}(2)\)-radius for that subclass is equal to \(\frac{2}{3}\). They conjectured that the same is valid for the class \({\mathcal {U}}\) [7]. In the second part of this paper, we shall prove that the conjecture is not true by giving the correct \({\mathcal {P}}(2)\)-radius for the class \({\mathcal {U}}\).
Let \(\Omega \) be the subset of \({\mathcal {A}}\) which consists of all functions f satisfying
It is known that \(\Omega \subset S^*\)[8]. In the third part of this paper, we shall give two conditions for functions to be in the class \(\Omega \).
2 \({\mathcal {P}}(2)\)-Radius for the Class \({\mathcal {U}}\)
Theorem 1
If \( f\in {\mathcal {U}}\), then
for \(|z|\le r_{0}=\frac{\sqrt{5}-1}{2}=0.618...\) and the result is the best possible.
For the proof of Theorem 1, we need the next lemma given by Shaffer[9].
Lemma 1
Let \(g(z)=\sum _{n=p}^{\infty }a_{n}z^{n} \,(p\ge 1)\) be analytic in \({{\mathbb {D}}}\) and satisfy \(|g(z)|\le 1\) for \( z\in {{\mathbb {D}}}\) , then
-
(a)
\(|g'(z)|\le p|z|^{p-1}\) for \(|z| \le \frac{\sqrt{1+p^{2}}-1}{p}\),
-
(b)
\(|g'(z)|\le |z|^{p-2}\,\frac{4|z|^{2}+p^{2}(1-|z|^{2})^{2}}{4(1-|z|^{2})}\) for \(|z|> \frac{\sqrt{1+p^{2}}-1}{p}\).
These estimates are the best possible.
Proof of Theorem 1
For \(f\in {\mathcal {U}}\) let’s put
Then,
If \(f(z)=z+a_{2}z^{2}+a_{3}z^{3}+....\), then
and
By using (1), previous notation and other conclusions, we can apply Lemma 1 with \(g(z)={\mathcal {U}}_{f}(z)\) and \(p=2\). By Lemma 1(a), we obtain
which by (4) implies
i.e., f has \({\mathcal {P}}(2)\)-property in the disk \(|z|\le r_{0}=\frac{\sqrt{5}-1}{2}\), which was to be proved. \(\square \)
Similarly, by Lemma 1(b) we have
where
It is easy to check that \(\varphi \) is an increasing function and \(\varphi (r_{0})=2<\varphi (t)\) for \(r_{0}<t<1\). For sharpness of the theorem , let us consider the function \(f_{b}\) defined by the condition
where b is real and \(|b|<1.\) Since \(\omega (z)=\frac{z+b}{1+b z}:{{\mathbb {D}}}\rightarrow {{\mathbb {D}}}\), then
which by (5) implies \(\frac{z}{f_{b}(z)}\ne 0,\, z\in {{\mathbb {D}}}\), i.e., \(f_{b}\) is well defined. Also
which gives that \(f_{b}\in {\mathcal {U}}\).
Let \(r_{1}\) be a fixed real number such that \(r_{0}<r_{1}<1\) and \(b_{1}=\frac{1-2r_{1}^2}{r^{3}_{1}}\). We claim that \(|b_{1}|<1\). In fact,
The left inequality is equivalent to \(r_{1}^{2}<1+r_{1}\), which is true, and the right is equivalent to \(1-r_{1}-r_{1}^{2}<0\), which is also true since \(r_{0}<r_{1}<1\).
After simple calculations, for the function \(f_{b_{1}}\) we have
because of the property of the function \(\varphi \) and since \(r_{0}<r_{1}<1\). It means that the function \(f_{b_{1}}\) is an extremal function for our problem, since it has \({\mathcal {P}}(2)\)-property in the disk \(|z|\le r_{0}=\frac{\sqrt{5}-1}{2}\) (because \(f_{b_{1}}\in {\mathcal {U}}\)), but not in a disk with longer radius.
3 Sufficient Conditions for Function to be in \(\Omega \)
Theorem 2
Let \( f\in {\mathcal {A}}\). If \(|f''(z)|\le 1\) then \(f\in \Omega \). The number 1 is the best possible.
Proof
Let \(g(z)=zf'(z)-f(z)\). Then, \(g'(z)=zf''(z)\). Since \(f(0)=f'(0)-1=0\) and \(|f''(z)|\le 1\) for \(z\in {\mathbb {D}}\), we have
where \(\omega (z)\in {{\mathcal {B}}}\). It follows from (6) that
Therefore,
That is, \(|zf'(z)-f(z)|<\frac{1}{2}\) for \(z\in {\mathbb {D}}\). This implies that \(f\in \Omega \subset S^*\).
If \(|f''(z)|\le \lambda \) and \(\lambda >1\), then f may be not univalent. For example, \(f(z)=z+\frac{1}{2}\lambda z^2 \) satisfy \(|f''(z)|\le \lambda \), but \(f'(z)=1+\lambda z\) vanish at \(-\frac{1}{\lambda }\), which implies that \(f\not \in S^*\). \(\square \)
Theorem 3
Let \( f\in {\mathcal {A}}\). If
then \(f\in \Omega \subset S^*\). The number \(\frac{3}{2}\) is the best possible.
Proof
Since \(f(0)=f'(0)-1=0\) and
it follows that
where \(\omega (z)\in {{\mathcal {B}}}\). Thus,
and consequently,
for \(z\in {\mathbb {D}}\). This implies that \(f\in \Omega \subset S^*\).
If \(|z^2f''(z)+zf'(z)-f(z)|\le \lambda \) and \(\lambda >\frac{3}{2}\), then f may be not univalent. One can see that by investigating the function \(f(z)=z+\frac{1}{2}\lambda z^2, \lambda >1 \). \(\square \)
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Communicated by See Keong Lee.
The work of the first author was supported by MNZZS Grant No. ON174017, Serbia. The research of the corresponding author was supported by the Key Laboratory of Applied Mathematics in Hubei Province, China.
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Obradović, M., Peng, Z. Some New Results for Certain Classes of Univalent Functions. Bull. Malays. Math. Sci. Soc. 41, 1623–1628 (2018). https://doi.org/10.1007/s40840-017-0546-0
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DOI: https://doi.org/10.1007/s40840-017-0546-0