UDC 517.54

MATHEMATICS

E. G. GOLUZINA

ON THE MUTUAL GROWTH OF THE COEFFICIENTS OF ONE CLASS OF \(p\)-VALENT FUNCTIONS

(Presented by Academician V. I. Smirnov, 19 XI 1965)

For the class \(S^{\square}\) of functions of the form \(f(z)=z+\sum_{n=2}^{\infty} a_n z^n\), regular and univalent in the mean with respect to area in the disk \(|z|<1\), Hayman \((^1)\) proved that

\[ \bigl||a_{n+1}|-|a_n|\bigr|<A,\qquad n\ge 2, \tag{1} \]

where \(A\) is an absolute constant. The order of estimate (1) is sharp.

For the class \(S^*\) of functions of the form \(f(z)=z+\sum_{n=2}^{\infty} a_n z^n\), univalent and starlike in the disk \(|z|<1\), estimate (1) (with \(A<100\)) was obtained by G. M. Goluzin \((^2)\) in 1946.

Hayman \((^3)\) posed the question whether

\[ \bigl||a_{n+1}|-|a_n|\bigr|\to 0 \quad \text{as } n\to\infty \]

for every function of the class \(S^{\square}\), other than functions of a certain special form.

In this direction, for the class \(S^*\) the following result of Pommerenke \((^4)\) is known: every function \(f(z)\in S^*\) either has the form

\[ \frac{z}{(1-e^{-i\theta_1}z)(1-e^{-i\theta_2}z)},\qquad \theta_1,\theta_2 \]

real, or there exists \(\delta=\delta(f)>0\) such that

\[ |a_{n+1}|-|a_n|=O(n^{-\delta}). \]

Lucas \((^3)\) generalized estimate (1) to the class of functions of the form \(f(z)=\sum_{n=0}^{\infty} a_n z^n\), regular and \(p\)-valent in the mean with respect to area \((p\ge 1)\) in the disk \(|z|<1\), proving that

\[ \bigl||a_{n+1}|-|a_n|\bigr|<A(p)\mu_p n^{2p-2},\qquad n\ge 1, \tag{2} \]

where \(\mu_p=\max_{0\le \nu\le p}|a_\nu|\), \(A(p)\) is a constant depending only on \(p\). The order of estimate (2) is sharp.

In the present paper, for a certain class of \(p\)-valent functions, a result analogous to the above-mentioned result of Pommerenke is obtained.

Let \(F(p)\) (\(p\) a fixed natural number) be the class of functions \(g(z)\) representable in the disk \(|z|<1\) by the formula

\[ g(z)=[\varphi(z)]^p z^{q-p}\prod_{s=1}^{p-q}\left(1-\frac{z}{\alpha_s}\right)(1-z\overline{\alpha}_s), \]

where \(q\) is an integer, \(1\le q\le p\), \(\varphi(z)\in S^*\), \(0<|\alpha_s|<1\), \(s=1,2,\ldots,p-q\); \(F(1)\equiv S^*\).

Let \(F(p,q)\) be the subclass of all functions from \(F(p)\) of the form

\[ g(z)=z^q+\sum_{n=q+1}^{\infty} a_n z^n \]

with fixed \(q\).

Bender \({}^{(5)}\) showed that all functions of the class \(F(p,q)\) are \(p\)-valent. For \(p>q\geqslant 1\), the class \(F(p,q)\) contains as a subclass the class \(S(p,q)\) \({}^{(5)}\) of functions of the form

\[ g(z)=z^q+\sum_{n=q+1}^{\infty} a_n z^n, \]

regular in the disk \(|z|<1\) and satisfying the condition: for each function \(g(z)\) there exists \(\rho\), \(0<\rho<1\), such that

\[ \operatorname{Re}\left[\frac{zg'(z)}{g(z)}\right]>0,\qquad \rho<|z|<1, \]

\[ \int_0^{2\pi}\operatorname{Re}\left[\frac{zg'(z)}{g(z)}\right]\,d\theta=2\pi p,\qquad z=re^{i\theta},\quad \rho<r<1. \]

For \(p=q\) we have \(F(p,p)\equiv S(p,p)\).

Theorem. If

\[ g(z)=z^q+\sum_{n=q+1}^{\infty} a_n z^n\in F(p,q) \]

and \(p>1\), and if

\[ g(z)\not\equiv \frac{z^q}{(1-e^{-i\theta}z)^{2p}} \prod_{s=1}^{p-q}\left(1-\frac{z}{\alpha_s}\right)(1-z\overline{\alpha}_s), \]

then there exists \(\delta=\delta(g)>0\) such that

\[ |a_{n+1}|-|a_n|=O\left(n^{2p-2-\delta}\right),\qquad n\geqslant q. \]

It is not difficult to show that there exist functions of the class \(F(p,q)\), of the form excluded in the theorem, for which

\[ |a_{n+1}|-|a_n|\sim K n^{2p-2}\quad \text{as } n\to\infty, \]

where \(K\) is a constant independent of \(n\).

In the proof of the theorem the following result is established:

Lemma. Let \(m\geqslant 1\), \(m\) an integer, and let

\[ g(z)=z^q+\sum_{n=q+1}^{\infty} a_n z^n\in F(p,q). \]

If

\[ g(z)\not\equiv z^q\prod_{k=1}^{m+1}(1-e^{-i\theta_k}z)^{-2p/(m+1)} \prod_{s=1}^{p-q}\left(1-\frac{z}{\alpha_s}\right)(1-z\overline{\alpha}_s) \]

(\(\theta_k\) real), then there exist complex numbers \(c_k\), \(k=0,1,\ldots,m\), \(|c_0|=|c_m|=1\), and \(\delta>0\), depending only on \(m\) and \(g\), such that for \(n\geqslant q\) we have

\[ \left|c_0 n^2 a_n+c_1(n+1)^2a_{n+1}+\cdots+c_m(n+m)^2a_{n+m}\right| =O\left(n^{2/(m+1)+2p-1-\delta}\right). \]

For \(p=1\), an analogous result was obtained by Pommerenke \({}^{(4)}\) for the class of close-to-convex functions.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
11 XI 1965

References

\({}^{1}\) W. K. Hayman, J. London Math. Soc., 38, pt. 2 (1963).
\({}^{2}\) G. M. Goluzin, Mat. sbornik, 19 (61), No. 2 (1946).
\({}^{3}\) U. Heil, Sborn. per. Matematika, 8, 1 (1964).
\({}^{4}\) Ch. Pommerenke, Proc. London Math. Soc., ser. 3, 13 (1963).
\({}^{5}\) Y. Bender, Duke Math. J., 29, No. 1 (1962).