Abstract
Jack’s Lemma says that if f(z) is regular in the disc \(|z|\le r\), \(f(0)=0\), and |f(z)| assumes its maximum at \(z_0\) on the circle \(|z|=r\), then \(z_0f'(z)_0/f(z_0)\ge 1\). This Lemma was generalized in several directions. In this paper we consider an improvement of some first author’s results of this type.
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1 Introduction
Let \(\mathcal {H}\) be the family of analytic functions in the region \( \mathbb {D}=\left\{ z:\left| z\right| <1\right\} \) on the complex plane \(\mathbb {C}\). Let \(\mathcal {A}\subset \mathcal {H}\) denote the set of all functions f(z) that are analytic in \(\mathbb {D}\) with the series representation
Also by \(\mathcal {S}\) we means a subfamily of the set \(\mathcal {A}\) which contains univalent functions. We next denote by \(\mathcal {P}\) the class of analytic functions p(z) which are normalized by
such that \(\mathfrak {Re} \{p(z)\} >0,\ \ z\in \mathbb {D}.\) Furthermore, by using the set \(\mathcal {P}\), let \(\mathcal {S}^{*}\) and \( \mathcal {C}\) denote the families of starlike and convex functions in \(\mathbb {D}\) which are defined as
The above subfamilies of the set \(\mathcal {S}\) are among the most studied families. Recall also, that if \(f\in \mathcal A\) satisfies
for some \(g\in \mathcal S^*\) and some \(\alpha \in (-\pi /2,\pi /2)\), then f is said to be close-to-convex in \(\mathbb D\) and denoted by \(f\in \mathcal K\). An univalent function \(f\in \mathcal A\) belongs to \(\mathcal K\) if and only if the complement E of the image-region \(F=\left\{ f(z): |z|<1\right\} \) is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays). We note that if \(g\left( z\right) =z\), then the class \(\mathcal {K}\) reduces to the set \(\mathcal {R}\) of bounded turning functions.
Jack’s Lemma [2], says that if f(z) is regular in the disc \(|z|\le r\), \(f(0)=0\), and |f(z)| assumes its maximum at \(z_0\) on the circle \(|z|=r\), then \(z_0f'(z)_0/f(z_0)\ge 1\). Now we will consider a lemma, which is a small extension of Jack’s Lemma.
We will test behavior of \(z_0f'(z)_0/f(z_0)\) when |f(z)| assumes its local maximum at \(z_0\) on the following neighborhood
Lemma 1.1
[1] Assume that \(p(z)=a_kz^k+a_{k+1}z^{k+1}+\cdots \), \(a_k\ne 0\), \(k\ge 1\) is analytic in \(\mathbb D\). If there exists \(z_0\in \mathbb D\) such that
for some \(\varepsilon >0\), then \(z_0p'(z_0)/p(z_0)\) is a real number and
Proof
Let us put
Then \(\varphi (z)\) is analytic in \(\mathbb D\) and \(|\varphi (z)|\) takes its local maximum value at the point \(z=z_0\) on the circular arc \(z=|z_0|e^{i\theta }\), \(\arg z_0-\varepsilon<\theta <\arg z_0+\varepsilon \). When z moves on the circle \(|z|=|z_0|\) with positive direction, then \(\arg \varphi (z)\) is increasing at \(z=z_0\). Therefore, we have
and so
From the hypothesis, we have
so
too. Therefore, we have
This shows
\(\square \)
Theorem 1.2
Assume that \(p(z)=1+a_kz^k+a_{k+1}z^{k+1}+\cdots \), \(a_k\ne 0\), \(k\ge 1\) is analytic in \(\mathbb D\). Assume there exist two points \(z_1,z_2\in \mathbb D\) such that \(|z_1|=|z_2|\) and
for some \(\alpha \), \(\beta \), \(\alpha ,\beta \in (0,2)\). If
and
for some \(\varepsilon >0\), then we have
where \(p^{1/\alpha }(z_1)=ia\), \(a>0\), and
where \(p^{1/\beta }(z_2)=-ib\), \(b>0\).
Proof
Let us check the first case \(\arg p(z_1)=\pi \alpha /2\) and put
then it follows that
and so
From the hypothesis, we have
for some \(\varepsilon >0\) and
Therefore, we have
and
This shows that \(|\varphi (z)|\) takes its local maximum value at the point \(z=z_1\) in the the domain \(N_\varepsilon (z_1)\) and applying Lemma 1.1, we have
where \(k\ge 1\) and \(q(z_1)=ia\) and \(a>0\). This shows that \(z_1q'(z_1)\) is a negative real number. This gives
On the other hand, we have
and putting \(z=|z_1|e^{i\theta }\) in the section \(\arg z_1-\varepsilon<\theta <\arg z_1+\varepsilon \), then \(\arg \{q(z_1)\}\) takes its local maximum \(\pi /2\) at the point \(\theta =\arg \{z_1\}\) in this section and so, we have
Therefore, we have
where \(p^{1/\alpha }(z_1)=ia\), \(a>0\). This completes the proof of (1.13).
For the next case \(\arg p(z_2)=-\pi \beta /2\) let us put
then it follows that
and so
From the hypothesis, we have
for some \(\varepsilon >0\) and
Therefore, we have
and
This shows that \(|\psi (z)|\) takes its local maximum value at the point \(z=z_2\) in the the domain \(N_\varepsilon (z_2)\) and applying Lemma 1.1, we have
where \(k\ge 1\) and \(\rho (z_2)=-ib\) with \(b>0\). This shows that \(z_2\rho '(z_2)\) is a negative real number. This gives
On the other hand, we have
and putting \(z=|z_2|e^{i\theta }\) in the section \(\arg z_2-\varepsilon<\theta <\arg z_2+\varepsilon \), then \(\arg \{\rho (z_2)\}\) takes its local minimum \(-\pi /2\) at the point \(\theta =\arg \{z_2\}\) in this section and so, we have
Therefore, we have
is a pure imaginary number, where \(p^{1/\beta }(z_2)=-ib\), \(b>0\) so
This completes the proof of (1.14). \(\square \)
Theorem 1.2 is a generalization of a result in [4, 5].
Corollary 1.3
Under the assumptions of Theorem 1.2 we have
Corollary 1.3 implies that the length of image curve under \(zp'(z)/p(z)\) of the circular arc \(|z|=|z_1|=|z_2|\) from \(z_2\) to \(z_1\) is great or equal to \((\alpha +\beta )k\).
Corollary 1.4
Under the assumptions of Theorem 1.2 and if \(\arg z_2<\arg z_1\) and
then we have
Proof
From Lemma 1.1 we have
On the other hand and applying (1.15), we have
where \(\arg z_1=\theta _1\), \(\arg z_2=\theta _2\) and \(\theta _2<\theta _1\). This shows that
which implies \( 0<\alpha <1\) and \( 0<\beta <1\) or
\(\square \)
In [8, p.54] we can find the following result for the convolution of power series
Lemma 1.5
If \(g(z)\in {\mathcal C}\), \(f(z)\in {\mathcal S}^*\) and F(z) is analytic in \(\mathbb D\), then for all \(z\in \mathbb D\)
where \(\overline{co}A\) denotes the closed convex hull of A.
Theorem 1.6
If \(f(z)=a_0+a_1z+a_2z^2+\cdots , h(z)=b_0+b_1z+b_2z^{2}+\cdots \in \mathcal H\), \(zh''(z)/(2b_2)\in \mathcal S^*\) and
are analytic functions in \(\mathbb D\), then
Proof
Denote
then \(s(z)\in \mathcal C\).
Applying Lemma 1.5, we have
because \(s(z)\in \mathcal C\) and \(zh''(z)/(2b_2)\in \mathcal S^*\). \(\square \)
The differential subordination theory provides another method od proving this type of results. For two functions f(z) and g(z) analytic in \(\mathbb D\), we say that ff(z) is subordinate to g(z), written by \(f\left( z\right) \prec g\left( z\right) \), if there exists a function w(z), analytic in \(\mathbb D\), with \(w(0) =0\) and \(|w(z)| <1\) such that \( f(z)=g( w(z)) \).
In [3, p.70] we can find the following result
Lemma 1.7
Let H(z) be convex univalent, \(P(z)\in \mathcal H\) and \(\mathfrak {Re}\{P(z)\}>0\) in \(\mathbb D\). If p(z) is amalytic in \(\mathbb D\) with \(p(0)=H(0)\), then
Theorem 1.8
Let H(z) be convex univalent in \(\mathbb D\). Assume that
are analytic in \(\mathbb D\) for some positive integer s and
If
and with \(f^{(s-1)}(0)/g^{(s-1)}(0)=H(0)\) then
Proof
Denote
Then we have \(p(z)h^{(s-1)}(z)=f^{(s-1)}(z)\), by differentation, we obtain
Applying (1.17) and (1.18) give \(p(z) \prec H(z)\), i. e. (1.19).
Corollary 1.9
Assume that \(\alpha \in (0,1]\) and
are analytic in \(\mathbb D\) Then
and
Consider, \(h(z)=e^z-1\), which satisfies
Therefore, \(h(z)\in \mathcal C\) and so \(h(z)\in \mathcal S^*\) which gives
Applying (1.20) we obtain
Corollary 1.10
Assume that \(\alpha \in [0,1)\) and
are analytic in \(\mathbb D\). Then
and
Property (1.22) was known earlier, see Theorem 2.6f in [3, p.63].
Theorem 1.11
Assume that \(h(z)\in \mathcal H\) and
for some \(\gamma \in (1,\infty ]\) and
for some \(\beta \in (0,1]\). Assume that \(f(z)\in \mathcal H\) and
for some \(\alpha \in (0,1]\). Furthermore, assume that
is analytic in \(\mathbb D\) with \(F(0)=1\). Then
Proof
Consider Theorem 1.8 with \(s=2\), \(H(z)=\left( \frac{1+z}{1-z}\right) ^{\alpha }\) and \(g'(z)= h'(z)-\gamma -z\). Then H(z) is convex univalent and condition (1.18) becomes (1.24). Applying this for (1.26) gives us
Now, applying (1.25), we have
because of (1.28). So we have
A simple geometric observation shows that
Therefore,
\(\square \)
Condition (1.27) implies that f(z) is strongly close-to-convex of order \(\alpha +\beta \) with respect convex function \(g(z)=z\). We obtain more when \(\alpha +\beta \le 1\) in the following corollary.
Corollary 1.12
Assume that \(h(z)\in \mathcal H\) and
for some \(\gamma \in (1,\infty ]\) and
for some \(\beta \in (0,1]\). Assume that \(f(z)\in \mathcal H\) and
for some \(\alpha \in (0,1-\beta ]\). Furthermore, assume that
is analytic in \(\mathbb D\) with \(F(0)=1\). Then
Proof
We have \(\alpha +\beta \le 1\), then by Theorem 1.11, we have
This gives (1.32). \(\square \)
Condition (1.32) implies the univelence f(z) in \(\mathbb D\) by Noshiro–Warshawski’s theorem.
Data Availability Statement
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Nunokawa, M., Sokół, J. On an extension of Nunokawa’s lemma. Anal.Math.Phys. 12, 36 (2022). https://doi.org/10.1007/s13324-021-00598-x
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DOI: https://doi.org/10.1007/s13324-021-00598-x