Abstract
We prove some new sufficient conditions for a function to be p-valent or p-valently starlike in the unit disc.
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1 Introduction
Let \(\mathcal{A}_{p}\) be the class of analytic functions of the form
A function \(f(z)\) which is analytic in a domain \(D \subset C\) is called p-valent in D if for every complex number w, the equation \(f(z)= w\) has at most p roots in D, and there will be a complex number \(w_{0}\) such that the equation \(f(z) = w_{0}\) has exactly p roots in D. Further, a function \(f\in\mathcal{A}_{p}\), \(p\in\mathbb{N}\setminus\{1\}\), is said to be p-valently starlike of order α, \(0\le\alpha< p\), if
The class of all such functions is usually denoted by \(\mathcal{S}^{*}_{p}(\alpha)\). For \(p=1\), we receive the well-known class of normalized starlike univalent functions \(\mathcal{S}^{*}(\alpha)\) of order α, \(\mathcal{S}_{p}^{*}(0)=\mathcal{S}_{p}^{*}\). If \(zf'(z)\in\mathcal{S}^{*}(\alpha)\), then \(f(z)\) is said to be p-valently convex of order α, \(0\le\alpha< p\). The class of all such functions is usually denoted by \(\mathcal{C}_{p}(\alpha)\). For \(p=1\), we receive the well-known class of normalized convex univalent functions \(\mathcal{C}(\alpha)\) of order α, \(\mathcal{C}_{p}(0)=\mathcal{C}_{p}\).
The well-known Noshiro-Warschawski univalence condition (see [1] and [2]) indicates that if \(f(z)\) is analytic in a convex domain \(D\subset\mathbb{C}\) and
where θ is a real number, then \(f(z)\) is univalent in D. In [3] Ozaki extended the above result by showing that if \(f(z)\) of the form (1.1) is analytic in a convex domain D and for some real θ we have
then \(f(z)\) is at most p-valent in D. Applying Ozaki’s theorem, we find that if \(f(z)\in\mathcal{A}_{p}\) and
then \(f(z)\) is at most p-valent in \(|z|<1\). In [4] it was proved that if \(f(z)\in\mathcal{A}_{p}\), \(p\geq2\), and
then \(f(z)\) is at most p-valent in \(|z|<1\).
2 Preliminary lemmata
Lemma 2.1
[5]
Let \(f(z) = z + a_{2} z^{2} + \cdots\) be analytic in the unit disc and suppose that
then \(f(z)\) is univalent in \(|z| < 1\).
Lemma 2.2
[6]
Let \(p(z)\) be an analytic function in \(|z|<1\) with \(p(0)=1\), \(p(z)\neq0\). If there exists a point \(z_{0}\), \(|z_{0}|<1\), such that
and
for some \(0<\alpha<2\), then we have
where
and
where
Lemma 2.3
[7]
Let p be a positive integer. If \(f(z)=z^{p}+\sum_{n=p+1}^{\infty}a_{n}z^{n}\) is analytic in \(\mathbb{D}\) and if it satisfies
then \(f(z)\) is p-valently starlike in \(\mathbb{D}\) and
for \(k=1,2,\ldots,(p-1)\).
Lemma 2.4
([7], p.282)
Let \(f\in\mathcal{A}_{p}\). If there exists a \((p-k+1)\)-valent starlike function \(g(z)=\sum_{n=p-k+1}^{\infty}b_{n}z^{n}\) (\(b_{p-k+1}\neq0\)) that satisfies
then \(f(z)\) is p-valent in \(|z|<1\).
3 Main results
Now we state and prove the main results. The first theorem poses a growth condition to a higher derivative of an analytic function. Hence, its proof, among others, uses the method of integrating the derivatives, which is frequently used in complex analysis, especially in estimating point evaluation operators (see, e.g., Lemma 7 in [8] and Lemma 4 in [9]).
Theorem 3.1
Let \(f\in\mathcal{A}_{p}\) and suppose that
where \(\beta_{0}=0.38\cdots\) is the positive root of the equation
and
Then we have
and, therefore, we have
or \(f(z)\) is p-valently starlike in \(|z| < 1\).
Proof
From the hypothesis (3.1), we have
This shows that
Let us put
Then it follows that
If there exists a point \(z_{0}\), \(|z_{0}|<1\), such that
and
then by Lemma 2.2 we have
where
and
For the case (3.7), we have
This contradicts (3.5), and for the case (3.8), applying the same method as above, we have
This also contradicts (3.5) and, therefore, it shows that
Applying (3.5) and (3.9), we have
This shows that
and by Lemma 2.3 we have
or \(f(z)\) is p-valently starlike in \(|z| < 1\). □
Remark
Note that if \(m(x)=2x+\frac{2}{\pi}\tan^{-1}x\), then
Hence, if
then \(\beta_{0}=0.38\cdots\). Moreover,
and
For \(p=1\), Theorem 3.1 becomes the following corollary which extends the result contained in Lemma 2.1.
Corollary 3.2
Let \(f\in\mathcal{A}(1)\) and suppose that
where \(\beta_{0}=0.38\cdots\) is the positive root of the equation
and
Then we have
or \(f(z)\) is starlike univalent in \(|z| < 1\).
An analytic function \(f(z)\) is said to be typically real if the inequality \(\mathfrak{Im}z\mathfrak{Im} f(z)\geq0\) holds for all \(z\in\mathbb{D}\). From the definition of a typically real function it follows that \(z\in\mathbb{D}^{+}\Leftrightarrow f(z)\in\mathbb{C}^{+}\) and \(z\in\mathbb{D}^{-}\Leftrightarrow f(z)\in\mathbb{C}^{-}\). The symbols \(\mathbb{D}^{+}\), \(\mathbb{D}^{-}\), \(\mathbb{C}^{+}\), \(\mathbb{C}^{-}\) stand for the following sets: the upper and the lower halves of the disk \(\mathbb{D}\), the upper half-plane and the lower half-plane, respectively.
Theorem 3.3
Let \(f(z)\in\mathcal{A}_{p}\) and suppose that
where \(g(z)\) is univalent starlike in \(\mathbb{D}\) and the functions
are typically real in \(\mathbb{D}\). Then we have
Proof
Let us put
Then it follows that
From the hypothesis
is typically real in \(\mathbb{D}\). If there exists a point \(z_{0}\), \(|z_{0}|<1\), such that
and
then by Lemma 2.2 we have
where
and
For the case
we have
hence
because \(zq'(z)/q(z)\) is typically real. Therefore, (3.19) yields that
because \(zg'(z)/g(z)\) is typically real. Moreover,
because \(g(z)\) is a univalent starlike function, see [10]. Applying (3.18), (3.20) and (3.21) in (3.16), we have
This contradicts (3.13), and for the case
applying the same method as above, we have
This also contradicts (3.13) and, therefore, it shows that (3.15) holds. □
Corollary 3.4
Let \(f(z)\in\mathcal{A}_{p}\) and all the coefficients of \(f(z)\) are real and suppose that
where \(g(z)\) is univalent starlike and typically real in \(\mathbb{D}\). Then we have
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Nunokawa, M., Sokół, J. On some geometric properties of multivalent functions. J Inequal Appl 2015, 300 (2015). https://doi.org/10.1186/s13660-015-0818-x
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DOI: https://doi.org/10.1186/s13660-015-0818-x