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MATHEMATICS
S. A. GEL'FER
ON THE COEFFICIENTS OF TYPICALLY REAL FUNCTIONS
(Presented by Academician V. I. Smirnov, 1 II 1957)
A function \(f(z)\) is called typically real in the disk \(|z|<1\) if it is real for real \(z\), and at the remaining points of this disk satisfies the condition
\[ \operatorname{Im}(f(z))\cdot \operatorname{Im}(z)>0. \tag{1} \]
Let \(T\) be the class of typically real functions \(f(z)\), regular in \(|z|<1\) and normalized by the conditions \(f(0)=0,\ f'(0)=1\). In the present article we consider the subclass \(T^{(2)}\) of functions \(f_2(z)\in T\) such that also \(\widetilde f_2(z)=\frac{1}{i}f_2(iz)\in T\) \((^1)\). These functions, in addition to condition (1), satisfy the condition
\[ \operatorname{Re}(f_2(z))\cdot \operatorname{Re}(z)>0, \tag{2} \]
and all of them are odd. Put
\[ f_2(z)=z+x_1z^3+x_2z^5+\cdots+x_nz^{2n+1}+\cdots . \tag{3} \]
We note that the class \(T^{(2)}\) contains all odd univalent functions of the form (3) with real coefficients.
For the coefficients \(x_n\) the following sharp estimates are known \((^{1,2})\):
\[ |x_n|+|x_{n-1}|\leqslant 2 \qquad (n=2,3,\ldots); \tag{4} \]
\[ -1\leqslant x_1\leqslant 1,\qquad -\frac{1}{2}\leqslant x_2\leqslant \frac{3}{2}. \tag{5} \]
There are no sharp estimates of the individual coefficients \(x_n\) for \(n>2\).
Theorem. If \(f_2(z)\in T^{(2)}\), then
\[ -1-\frac{\sqrt{3}}{18}\leqslant x_3\leqslant 1+\frac{\sqrt{3}}{18}=1.09\ldots; \tag{6} \]
\[ x_{2k}\leqslant \frac{3}{2}\qquad (k=1,2,\ldots); \tag{7} \]
\[ x_4\geqslant -\frac{2}{3}; \tag{8} \]
\[ x_6\geqslant -\frac{5}{16}-\frac{121}{48\sqrt{33}}\simeq -\frac{3}{4}. \tag{9} \]
The estimates are sharp, and each bound is attained by its own unique function belonging to the class \(T^{(2)}\).
Proof. Let \(f_2(z)\in T^{(2)}\). Put
\[ \varphi(z)=\frac{1-z^2}{z}\,f_2(z^{1/2})=1+\alpha_1z+\cdots+\alpha_nz^n+\cdots . \tag{10} \]
The relations hold
\[ \alpha_n= \left( 2x_n+2x_1x_{n-1}+\cdots+ \begin{matrix} \nearrow x_{n/2}^{\,2}\\ \searrow 2x_{(n-1)/2}x_{(n+1)/2} \end{matrix} \right) - \left( 2x_{n-2}+2x_1x_{n-3}+\cdots+ \begin{matrix} \nearrow x_{n/2-1}^{\,2}\\ \searrow 2x_{(n-3)/2}x_{(n-1)/2} \end{matrix} \right) \begin{matrix} (n\ \text{even})\\ (n\ \text{odd}) \end{matrix} \tag{11} \]
\[ (n=1,2,\ldots;\quad x_0=1;\quad x_n=0,\ \text{if }x<0). \]
The function (10) is regular in the disk \(|z|<1\), satisfies there the condition \(\operatorname{Re}(\varphi(z))>0\) \((^1,{}^2)\), and all coefficients \(\alpha_n\) are real. Conversely, to every function \(\varphi(z)\) possessing these properties there corresponds, by (10), a function \(f_2(z)\in T^{(2)}\). We shall denote the class of such functions \(\varphi(z)\) by \(R\).
Put
\[ \delta_n= \left| \begin{array}{ccccc} 2 & \alpha_1 & \alpha_2 & \cdots & \alpha_n\\ \alpha_1 & 2 & \alpha_1 & \cdots & \alpha_{n-1}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \alpha_n & \alpha_{n-1} & \alpha_{n-2} & \cdots & 2 \end{array} \right| \qquad (n=1,2,\ldots). \tag{12} \]
By Carathéodory’s theorem \((^3)\), the conditions \(\delta_n\geqslant0\) \((n=1,2,\ldots)\) are necessary and sufficient for \(\varphi(z)\in R\); moreover, if \(\delta_{n_0}=0\), then for \(n>n_0\) all \(\delta_n=0\). Hence, taking into account the relations (11), one can obtain estimates for \(x_n\) in terms of \(x_1,\ldots,x_{n-1}\). Let, for definiteness, \(n=2k\). Introduce the notation
\[ \Delta_1^{(k)}=\Delta(\alpha_1,\ldots,\alpha_{2k})= \left| \begin{array}{cccc} 2-\alpha_{2k} & \alpha_1-\alpha_{2k-1} & \cdots & \alpha_{k-1}-\alpha_{k+1}\\ \alpha_1-\alpha_{2k-1} & 2-\alpha_{2k-2} & \cdots & \alpha_{k-2}-\alpha_k\\ \cdot & \cdot & \cdot & \cdot\\ \alpha_{k-1}-\alpha_{k+1} & \alpha_{k-2}-\alpha_k & \cdots & 2-\alpha_2 \end{array} \right| \qquad (k=1,2,\ldots); \tag{13} \]
\[ \Delta_2^{(k)}=\Delta_2(\alpha_1,\ldots,\alpha_{2k})= \left| \begin{array}{cccc} 1 & \alpha_1 & \cdots & \alpha_k\\ \alpha_1 & 2+\alpha_2 & \cdots & \alpha_{k-1}+\alpha_{k+1}\\ \cdot & \cdot & \cdot & \cdot\\ \alpha_k & \alpha_{k-1}+\alpha_{k+1} & \cdots & 2+\alpha_{2k} \end{array} \right|. \tag{14} \]
Performing elementary transformations of the determinant (12) \((n=2k)\), we obtain \(\delta_{2k}=\Delta_1^{(k)}\cdot\Delta_2^{(k)}\), whence, on the basis of Carathéodory’s conditions,
\[ \frac{ \left| \begin{array}{cccc} 1 & \alpha_1 & \cdots & \alpha_k\\ \alpha_1 & 2+\alpha_2 & \cdots & \alpha_{k-1}+\alpha_{k+1}\\ \cdot & \cdot & \cdot & \cdot\\ \alpha_k & \alpha_{k-1}+\alpha_{k+1} & \cdots & 2 \end{array} \right| }{ \Delta_2(\alpha_1,\ldots,\alpha_{2k-2}) } \leqslant \alpha_{2k} \leqslant \frac{ \left| \begin{array}{cccc} 2 & \alpha_1-\alpha_{2k-1} & \cdots & \alpha_{k-1}-\alpha_{k+1}\\ \alpha_1-\alpha_{2k-1} & 2-\alpha_{2k-2} & \cdots & \alpha_{k-2}-\alpha_k\\ \cdot & \cdot & \cdot & \cdot\\ \alpha_{k-1}-\alpha_{k+1} & \alpha_{k-2}-\alpha_k & \cdots & 2-\alpha_2 \end{array} \right| }{ \Delta_1(\alpha_1,\ldots,\alpha_{2k-2}) } \tag{15} \]
\[ (k=1,2,\ldots;\quad \Delta_1^{(0)}=\Delta_2^{(0)}=1). \]
Denote by \(A_1(x_1,\ldots,x_{2k-1})\) and \(A_2(x_1,\ldots,x_{2k-1})\) the functions of \(2k-1\) variables \(x_1,\ldots,x_{2k-1}\) which are obtained as a result of the substitution
respectively into the right- and left-hand inequalities (15), instead of \(\alpha_1,\ldots,\alpha_{2k-1}\), their expressions (11) in terms of \(x_1,\ldots,x_{2k-1}\); we obtain
\[ A_2(x_1,\ldots,x_{2k-1})-B(x_1,\ldots,x_{2k-1})\leq 2x_{2k}\leq \tag{16} \]
\[ \leq A_1(x_1,\ldots,x_{2k-1})-B(x_1,\ldots,x_{2k-1}), \]
where
\[ B(x_1,\ldots,x_{2k-1})= \]
\[ =(2x_1x_{2k-1}+2x_2x_{2k-2}+\cdots+x_k^2) -(2x_{2k-2}+2x_1x_{2k-3}+\cdots+x_{k-1}^2). \]
Analogous inequalities can also be written for \(2x_{2k+1}\).
We shall assign to each series (3) a point \(x\) with Cartesian coordinates \(x_1,\ldots,x_n\) in the \(n\)-dimensional Euclidean space \(R_n\) \((n\geq 1)\). To the class \(T^{(2)}\) there will correspond in \(R_n\) a closed convex body \(T_n^{(2)}\), since, if \(f_2^{(1)}(z)\in T\) and \(f_2^{(2)}(z)\in T^{(2)}\), then also \(\lambda f_2^{(1)}(z)+(1-\lambda)f_2^{(2)}(z)\in T^{(2)}\) for any \(\lambda,\ 0\leq\lambda\leq1\). Taking the equality sign in relations (16), we obtain the equation of the part \(T_{n-1}^{(2)}\) of the boundary of this body that is convex in the direction of the positive, respectively negative, axis \(x_n\).
If within \(T_{n-1}^{(2)}\) there exist stationary points of the functions
\(A_i(x_1,\ldots,x_{2k-1})-B(x_1,\ldots,x_{2k-1})\) \((i=1,2)\), then their values at these points, in view of the convexity of \(T_n^{(2)}\), will respectively be the upper and lower exact bounds of \(2x_n\). In this way, for \(n=1,2\) we find the estimates (5), and for \(n=3\) we obtain the estimates (6), the equality sign on the right in (6) being attained for
\[ x_1=1-\frac1{\sqrt3},\quad x_2=1-\frac1{2\sqrt3}, \]
and on the left for
\[ x_1=-1+\frac1{\sqrt3},\quad x_2=1-\frac1{2\sqrt3}. \]
The uniqueness of the extremal functions follows from the fact that at these stationary points there is a strict extremum. Determination of the stationary points and their investigation for arbitrary \(n\) is very difficult.
To obtain the estimates (7)—(9), consider the function
\[ \psi_2(z)=\frac{f_2(z)+\overline{f_2(\overline z)}}{2} =z+x_2z^5+\cdots+x_{2n}z^{4n+1}+\cdots \tag{17} \]
It is not hard to see that \(\psi_2(z)\in T^{(2)}\). To the class of functions \(\psi_2(z)\) defined by formula (17) there will correspond a closed and convex body \(\widetilde T_n^{(2)}\). Let, for definiteness, \(n=2k\). For the part of the boundary of this body convex in the direction of the positive axis \(x_{2n}\), we have the equation
\[ 2x_{4k}=A_1(x_2,x_4,\ldots,x_{4k-2})- \]
\[ -(2x_2x_{4k-2}+2x_4x_{4k-4}+\cdots+x_{2k}^2) +(2x_{4k-2}+2x_2x_{4k-4}+\cdots+2x_{2k-2}x_{2k}). \tag{18} \]
It is not hard to notice that the right-hand side of (18) has the stationary point
\(x_2=x_4=\cdots=x_{4k-2}=1\), lying inside \(\widetilde T_{n-1}^{(2)}\). The corresponding values are \(\alpha_n=1\) \((n=1,2,\ldots,2k-1)\), whence we find \(x_{4k}=3/2\). The same is obtained also for \(n=2k+1\). Hence the estimates (7) are obtained.
The uniqueness of the corresponding extremal functions follows from the fact that the right-hand side of (18) has a maximum at this stationary point.
For \(n=2,3\), by the same method we find exact estimates also from below. Estimate (8) is attained for \(x_2=1/3\), estimate (9) is attained for
\[ x_2=\frac{\sqrt{33}-3}{6},\quad x_4=\frac{9-\sqrt{33}}{24}. \]
The theorem is proved.
Gorky Civil Engineering Institute
named after V. P. Chkalov
Received
30 I 1957
References
- G. M. Goluzin, Matem. sborn., 27 (69): 2, 201 (1950).
- W. Rogosinski, Math. Zs., 35, 93 (1932).
- C. Carathéodory, Rend. Circ. Math. Palermo, 32, 193 (1911).