MATHEMATICS

I. M. MILIN

ESTIMATE OF THE COEFFICIENTS OF UNIVALENT FUNCTIONS

(Presented by Academician M. A. Lavrent’ev, October 3, 1964)

Let, as usual, \(S\) be the class of functions
\(f(z)=z+c_2z^2+\cdots\), regular and univalent in the disk \(|z|<1\), and let \(\Sigma\) be the class of functions
\(F(z)=z+\alpha_0+\alpha_1/z+\cdots\), regular and univalent in the domain \(1<|z|<\infty\).

Each function \(F(z)\in\Sigma\) generates a system of functions \(\{A_n(z)\}\) \((n=1,2,\ldots)\) by means of the expansion

\[ \ln \frac{z-t}{F(z)-F(t)} = \sum_{n=1}^{\infty} A_n(t) z^{-n}, \qquad |z|>1,\quad |t|>1 \tag{1} \]

(the branch of the logarithmic function is taken which tends to zero as \(z=\infty\)). In the author’s paper \((^1)\) the basic properties of the system \(\{A_n(z)\}\) are set forth and, in particular, for every \(z\) in the domain \(|z|>1\) the sharp inequality is established:

\[ \sum_{n=1}^{\infty} n |A_n(z)|^2 \le \ln \frac{1}{1-r^2}, \qquad |z|=\rho=\frac{1}{r}>1. \tag{2} \]

Using this inequality, it is possible to improve the estimate of the coefficients of univalent functions of the class \(S\). Below we give the derivation of a new estimate.

Theorem 1. For a function
\[ f(z)=z+\sum_{n=2}^{\infty} c_n z^n \in S \]
and for every \(\rho\in[1,\infty)\), the inequality

\[ |c_n| \le \rho^{\,n-1} \max_{|t|=\rho} \left| \left\{ \exp\left[ \sum_{k=1}^{n-1} A_k(t) z^{-k} \right] \right\}_{z=t}^{(n-1)} \right|, \qquad n=2,3,\ldots, \tag{3} \]

holds, where the system \(\{A_k(z)\}\) \((k=1,2,\ldots)\) is generated by the function

\[ F(z)=\frac{1}{f(1/z)}\in\Sigma,\qquad |z|>1, \tag{4} \]

and the symbol \(\{\psi(z)\}_{z=t}^{(n)}\) denotes the \(n\)-th partial sum of the Taylor expansion of the function \(\psi(z)\) about \(z=\infty\) at \(z=t\).

Proof. Let \(f(z)\) be an arbitrary function of the class \(S\), and let \(F(z)\) be the function of the class \(\Sigma\) corresponding to it by (4). Exponentiating equality (1), we introduce the system of functions \(\{D_n(z)\}\) \((n=1,2,\ldots)\):

\[ \frac{z-t}{F(z)-F(t)} = \exp\left[ \sum_{n=1}^{\infty} A_n(t) z^{-n} \right] = 1+\sum_{n=1}^{\infty} D_n(t) z^{-n}, \qquad |z|>1,\quad |t|>1. \tag{5} \]

Now from (5), by multiplying by \(z/(z-t)\), we obtain the expansion

\[ \frac{z}{F(z)-F(t)}=1+\sum_{n=1}^{\infty}\varepsilon_n(t)z^{-n},\qquad |z|>|t|>1, \tag{6} \]

where it is denoted

\[ \varepsilon_n(t)=t^n\sum_{k=0}^{n}D_k(t)t^{-k},\qquad n=1,2,\ldots;\qquad D_0=1,\quad |t|>1. \tag{7} \]

Alongside (6), for any finite \(w\) consider the function \(z/[F(z)-w]\) and its Taylor expansion about \(z=\infty\). We have:

\[ \frac{z}{F(z)-w}=1+\sum_{n=1}^{\infty}P_n(w)z^{-n}. \tag{8} \]

It is clear that the coefficients \(P_n(w)\) \((n=1,2,\ldots)\) are polynomials of degree \(n\) in \(w\).

If in equality (8) we put \(w=F(t)\), where \(t\) is an arbitrary finite value from the exterior of the unit circle, then, by uniqueness of the expansion of a regular function in a Taylor series, from (6) and (8) we obtain

\[ \varepsilon_n(t)=P_n(F(t)),\qquad n=1,2,\ldots,\quad |t|>1. \tag{9} \]

If, however, in equality (8) we put \(w=0\), then for the same reason, from (8) and (4) we shall have:

\[ c_{n+1}=P_n(0)\qquad (n=1,2,\ldots). \tag{10} \]

Since the function \(F(z)\) does not vanish in the domain \(|z|>1\), the interior of any level line \(C_\rho\) (\(C_\rho\) is the image of the circle \(|z|=\rho>1\) under the mapping by the function \(F(z)\)), \(1<\rho<\infty\), contains the point \(w=0\), and then, by the maximum-modulus principle for analytic functions, from (9) and (10), replacing \(n\) by \(n-1\), we derive the inequality

\[ |c_n|\leq \max_{w\in C_\rho}|P_{n-1}(w)|=\max_{|t|=\rho}|\varepsilon_{n-1}(t)|,\qquad n=2,3,\ldots,\quad 1\leq \rho<\infty \tag{11} \]

(the equality sign is possible only for \(\rho=1\); in this case the right-hand side of (11) is understood as the limit as \(\rho\to 1\)).

Inequality (11), taking (7) into account, can be rewritten as

\[ |c_n|\leq \rho^{\,n-1}\max_{|t|=\rho}\left|\sum_{k=0}^{n-1}D_k(t)t^{-k}\right|,\qquad n=2,3,\ldots,\quad 1\leq \rho<\infty,\quad D_0=1, \tag{12} \]

and in this form it coincides with inequality (3). The theorem is proved.

Inequalities (3) and (2) already make it possible to estimate \(|c_n|\) for all \(n\) \((n=2,3,\ldots)\), additionally using only Bunyakovsky’s inequality. To obtain a better estimate, we state in the form of a theorem (without proof) one more inequality, obtained by the author jointly with N. A. Lebedev.

Theorem 2. For any sequence of complex numbers \(\{A_k\}\) \((k=1,2,\ldots)\), for which

\[ \sum_{k=1}^{\infty} k|A_k|^2<\infty, \]

the inequality

\[ \sum_{k=0}^{\infty}|D_k|^2\leq \exp\left[\sum_{k=1}^{\infty} k|A_k|^2\right], \tag{13} \]

holds.

where it is denoted:

\[ \exp \left[\sum_{k=1}^{\infty} A_k z^{-k}\right] = \sum_{k=0}^{\infty} D_k z^{-k}, \qquad |z|>1. \]

The equality sign occurs if and only if \(A_k=(1/k)c^k\) \((k=1,2,\ldots)\), \(|c|<1\).

We now formulate the final result.

Theorem 3. For a function \(f(z)=z+\sum_{n=2}^{\infty} c_n z^n \in S\), the estimate

\[ |c_n|<1.243n \qquad (n=2,3,\ldots). \tag{14} \]

holds.

Proof. Estimating the modulus of the sum in (12) with the aid of Bunyakovsky’s inequality, we shall have:

\[ |c_n| \le \frac{1}{r^{\,n-1}} \max_{|t|=\rho} \left[ \sum_{k=0}^{n-1} |D_k(t)|^2 \sum_{k=0}^{n-1} r^{2k} \right]^{1/2}, \qquad n=2,3,\ldots,\quad |t|=\rho=\frac{1}{r}>1. \tag{15} \]

But from inequalities (2) and (13), taking (5) into account, we obtain:

\[ \sum_{k=0}^{\infty} |D_k(t)|^2 \le \frac{1}{1-r^2}, \qquad |t|=\rho=\frac{1}{r}>1. \tag{16} \]

Further, the joint consideration of inequalities (15) and (16) leads to the relation

\[ |c_n| < \frac{\sqrt{1-r^{2n}}}{r^{\,n-1}(1-r^2)}, \qquad n=2,3,\ldots,\quad 0<r<1. \tag{17} \]

Putting \(r^{2n}=e^{-x}\), \(0<x<\infty\), we express the first part in (17) in terms of \(x\). We have:

\[ |c_n| < \frac{\sqrt{e^x-1}}{x}\, \frac{x/n}{e^{x/2n}-e^{-x/2n}}\, n, \qquad n=2,3,\ldots,\quad 0<x<\infty. \tag{18} \]

Using for \(x\in(0,\infty)\) the obvious inequality

\[ \frac{x/n}{e^{x/2n}-e^{-x/2n}}<1 \qquad (n=2,3,\ldots), \]

from (18) we obtain for \(|c_n|\) the estimate:

\[ |c_n|< \frac{\sqrt{e^x-1}}{x}\,n, \qquad n=2,3,\ldots,\quad 0<x<\infty. \]

Choosing \(x=1.6\), we arrive at the conclusion of the theorem. The theorem is proved.

If, in Littlewood’s well-known method \({}^{2}\), in the estimates for \(|c_n|\) already at the first step—the transition from the modulus of the integral to the integral of the modulus—an excessive factor is imposed (for the Koebe function equal to \(e/2\)), then inequality (3) does not have this drawback.

Received
26 IX 1964

References

\({}^{1}\) I. M. Milin, DAN, 154, No. 2, 264 (1964). \({}^{2}\) J. E. Littlewood, London, Math. Soc., Ser. 2, 23, 481 (1925).