Abstract Generated abstract
This paper studies coefficient bounds for the subclass of normalized typically real functions in the unit disk that remain typically real under rotation by a factor of i, a class consisting of odd functions with real coefficients. Using an associated Carathéodory function with positive real part, determinant positivity conditions, and convexity properties of the coefficient bodies, the paper derives sharp estimates for several individual coefficients. It proves exact upper and lower bounds for the third coefficient, a sharp general upper bound of 3/2 for all even-indexed coefficients, and sharp lower bounds for the fourth and sixth coefficients, with uniqueness of the extremal functions in each stated case.
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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
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- W. Rogosinski, Math. Zs., 35, 93 (1932).
- C. Carathéodory, Rend. Circ. Math. Palermo, 32, 193 (1911).