MATHEMATICS

V. M. TERPIGOREVA

ON EXTREMAL PROBLEMS FOR ORLICZ CLASSES OF ANALYTIC FUNCTIONS IN THE UNIT DISK

(Presented by Academician V. I. Smirnov, 9 VIII 1961)

§ 1. Numerous works have been devoted to the question of studying extremal problems for analytic functions of the classes \(H_p\) (for the definition of these classes see \((^1)\)) (see, for example, \((^{2-7})\), where the preceding literature is also indicated). In the present paper extremal problems are investigated for classes of analytic functions with the Orlicz metric. The known results for the classes \(H_p\) are special cases of the theorems given below.

We shall use the notation and definitions of the book \((^8)\). In particular, we shall denote by \(p(t)\) and \(q(s)\) the right inverse functions;

\[ M(u)=\int_0^{|u|} p(t)\,dt \quad\text{and}\quad N(v)=\int_0^v q(s)\,ds \]

the mutually complementary \(N\)-functions; \(L_M^*\) the Orlicz space; \(E_M^*\) the closure in \(L_M^*\) of the set of bounded functions; \(\|f\|_M\) the Orlicz norm of an element \(f\in L_M^*\), and \(\|f\|_{(M)}\) the Luxemburg norm.

First we define the classes in which the extremal problems will be solved. Let \(f(z)\) be analytic in the disk \(|z|<1\). Put

\[ \|f\|_{(M)}^r=\inf k \tag{1} \]

over all \(k>0\) for which

\[ \int_0^{2\pi} M\left[\frac{|f(re^{i\theta})|}{k}\right]\,d\theta \le 1, \]

and

\[ \|f\|_M^r=\sup \int_0^{2\pi} |f(re^{i\theta})\,v(e^{i\theta})|\,d\theta, \qquad \int_0^{2\pi} N[v(e^{i\theta})]\,d\theta \le 1. \tag{2} \]

It is easy to show that \(\|f\|_{(M)}^r\) and \(\|f\|_M^r\) are increasing functions of \(r\). Denote by \(H_M^*\) the class of functions analytic in \(|z|<1\) for which

\[ \sup_r \|f\|_M^r < +\infty . \]

For \(f\in H_M^*\) put

\[ \|f(z)\|_M=\|f\|_M=\sup_r \|f\|_M^r = \lim_{r\to 1}\|f\|_M^r; \tag{3} \]

\[ \|f(z)\|_{(M)}=\|f\|_{(M)}=\sup_r \|f\|_{(M)}^r = \lim_{r\to 1}\|f\|_{(M)}^r . \tag{4} \]

If in \(H_M^*\) one introduces the norm defined by equality (3) or by equality (4), then \(H_M^*\) becomes a Banach space; moreover, the norms \(\|f\|_M\) (the Orlicz norm) and \(\|f\|_{(M)}\) (the Luxemburg norm) are equivalent to each other. Obviously, \(H_M^*\) is contained in \(H_1\); therefore every function \(f(z)\in H_M^*\) has angular boundary values almost everywhere. In what follows we shall sometimes identify a function \(f(z)\in H_M^*\) with its angular boundary ...

values \(f(e^{i\theta})\). Under this condition, with the aid of the results of paper \((^9)\), the following theorem can be obtained:

Theorem 1. \(H_M^*=H_1\cap L_M^*\), and in this case
\[ \|f(e^{i\theta})\|_M=\|f(z)\|_M=\|f\|_M, \]
\[ \|f(e^{i\theta})\|_{(M)}=\|f(z)\|_{(M)}=\|f\|_{(M)}, \]
where \(\|f(e^{i\theta})\|_M\bigl(\|f(e^{i\theta})\|_{(M)}\bigr)\) denotes the Orlicz (Luxemburg) norm of the boundary values \(f(e^{i\theta})\) of the function \(f(z)\).

By \(EH_M\) we shall denote the closure in the norm in \(H_M^*\) of the set of bounded analytic functions.

Theorem 2. \(EH_M=H_1\cap E_M\).

Denote by \(H_M^{*1}\) the unit sphere in the space \(H_M^*\) with norm (3), and by \(H_{(M)}^{*1}\) the unit sphere in the same space with norm (4).

Theorem 3. \(H_M^{*1}\), \(H_{(M)}^{*1}\) are compact in themselves in the sense of convergence inside \(|z|<1\).

Theorem 4. In order that a sequence \(\{f_n\}\in H_M^*\) converge \(E_N\)-weakly, it is necessary and sufficient that the norms \(\|f_n\|_M\) be uniformly bounded and the sequence \(\{f_n(z)\}\) converge uniformly inside the disk \(|z|<1\).

§ 2. We now pass to the main question: the study of extremal problems in our classes.

Theorem 5. For any \(v(e^{i\theta})\in L_N^*\) we have
\[ \sup_{\substack{f\in EH_M\\ \|f\|_{(M)}\le 1}} \left| \int_{0}^{2\pi} f(e^{i\theta})\,v(e^{i\theta})\,d\theta \right| = \sup_{\substack{f\in H_{(M)}^{*1}}} \left| \int_{0}^{2\pi} f(e^{i\theta})\,v(e^{i\theta})\,d\theta \right| = \inf_{\varphi\in H_N^*}\|v-\varphi\|_N, \tag{5} \]
and there exists an extremal function \(\varphi^*(e^{i\theta})\in H_N^*\).

Theorem 6. If \(v(e^{i\theta})\in E_N\), then there exists an extremal function in the left-hand side of equality (5). If \(q(s)\) is a continuous function, then the extremal functions \(f^*(e^{i\theta})\) and \(\varphi^*(e^{i\theta})\) are connected by the relations:
\[ f^*(e^{i\theta})\,[v(e^{i\theta})-\varphi^*(e^{i\theta})]\,e^{i\theta} = e^{i\alpha}q\!\left[k^*|v(e^{i\theta})-\varphi^*(e^{i\theta})|\right]\, |v(e^{i\theta})-\varphi^*(e^{i\theta})| \tag{6} \]
or
\[ f^*(e^{i\theta})[v(e^{i\theta})-\varphi^*(e^{i\theta})]e^{i\theta} = e^{i\alpha}\frac{1}{k^*}\, |f^*(e^{i\theta})|\, p[|f^*(e^{i\theta})|], \tag{7} \]
where also \(\|f^*\|_{(M)}=1\) and, moreover,
\[ \int_{0}^{2\pi} M[|f^*(e^{i\theta})|]\,d\theta=1. \]
Here \(\alpha\) is a real number, and \(k^*>0\) satisfies the condition
\[ \|v-\varphi^*\|_N = \frac{1}{k^*} \left\{ 1+\int_{0}^{2\pi}N\!\left[k^*|v(e^{i\theta})-\varphi^*(e^{i\theta})|\right]\,d\theta \right\}. \]

(The existence of such a \(k^*\) for continuous \(q(s)\) follows from \((^8)\), pp. 106–110.)

Conversely, if for two functions \(f^*(e^{i\theta})\in H_M^*\) and \(\varphi^*(e^{i\theta})\in H_N^*\) the relations (6) or (7) hold and
\[ \int_{0}^{2\pi} M[|f^*(e^{i\theta})|]\,d\theta=1, \]
then \(f^*(e^{i\theta})\) and \(\varphi^*(e^{i\theta})\) are extremal functions in equality (5).

The extremal function \(\varphi^*(z)\) is always unique, while the extremal function \(f^*(z)\) is unique up to a factor \(e^{i\alpha}\).

Theorem 7. If \(v(z)\) is a rational function and \(\beta_1,\ldots,\beta_n\) are all its poles inside \(|z|<1\), then
\[ \Phi(z)=zf^*(z)[v(z)-\varphi^*(z)] = \frac{A}{K^*}\,z\, \frac{\displaystyle\prod_{1}^{n-1}(z-\alpha_i)(1-\overline{\alpha_i}z)} {\displaystyle\prod_{1}^{n}(z-\beta_i)(1-\overline{\beta_i}z)}\,; \tag{8} \]

\[ f^*(z)=\prod'\frac{z-\alpha_i}{1-\overline{\alpha_i}z} \exp\left[ \frac1{2\pi}\int_0^{2\pi} \ln\left\{ S^{-1}\left[ |A|\frac{\prod_1^{n-1}|e^{i\theta}-\alpha_i|^2} {\prod_1^n|e^{i\theta}-\beta_i|^2} \right]\right\} \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\theta \right]; \tag{9} \]

\[ v(z)-\varphi^*(z)=\frac{A}{K^*}\prod''\frac{z-\alpha_i}{1-\overline{\alpha_i}z} \prod_1^{n-1}(1-\overline{\alpha_i}z)^2 \prod_1^n\left[(z-\beta_i)(1-\overline{\beta_i}z)\right]^{-1}\times \]

\[ {}\times \exp\left[ \frac1{2\pi}\int_0^{2\pi} \ln\left\{ S^{-1}\left[ |A|\frac{\prod_1^{n-1}|e^{i\theta}-\alpha_i|^2} {\prod_1^n|e^{i\theta}-\beta_i|^2} \right]\right\} \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\theta \right], \tag{10} \]

where \(\alpha_i\) \((|\alpha_i|<1)\), \(i=1,\ldots,n-1\), are certain points; \(\prod'\) extends over those \(\alpha_i\) which serve as zeros of \(f^*(z)\), and \(\prod''\) over those \(\alpha_i\) which serve as zeros of \(v(z)-\varphi^*(z)\); \(S^{-1}(\omega)\) is the function inverse to the function \(s(\omega)=\omega p(\omega)\) \((\omega\geq 0)\); \(A\) satisfies the condition

\[ \int_0^{2\pi} M\left\{ S^{-1}\left[ |A|\frac{\prod_1^{n-1}|e^{i\theta}-\alpha_i|^2} {\prod_1^n|e^{i\theta}-\beta_i|^2} \right]\right\}\,d\theta=1. \tag{11} \]

From formulas (8), (9) one immediately obtains the form of the extremal functions for the classes \(H_p\) \((p\geq 1)\) and for the class of bounded functions (see \((2\text{--}7)\)). Indeed, for \(H_p\), \(p\geq 1\), we have \(S(\omega)=\omega^p\), and formula (9) gives, for example,

\[ f^*(z)=C\prod'\frac{z-\alpha_i}{1-\overline{\alpha_i}z} \prod_1^{n-1}(1-\overline{\alpha_i}z)\cdot \prod_1^n(1-\overline{\beta_i}z)^{-2/p}. \]

§ 3. Applying the methods developed in the works \((^{10,\,11})\), one can investigate, by means of Theorem 7, the form of the extremal functions in problems of Carathéodory—Fejér—Nevanlinna—Pick type for our classes of functions.

Theorem 8. Let linear functionals \(l_1(f),\ldots,l_m(f)\) of the form

\[ l_i(f)=\int_0^{2\pi} f(e^{i\theta})\,v_i(e^{i\theta})\,d\theta, \qquad v_i(e^{i\theta})\in E_N,\quad i=1,\ldots,m, \]

and arbitrary complex numbers \(c_1,\ldots,c_m\) be given. Among the functions of the class \(H_M^*\) for which \(l_i(f)=c_i\), \(i=1,\ldots,m\), the smallest norm \(\|f\|_{(M)}\) is attained by a unique function \(f^*(z)\), such that \(f(z)=f^*(z)\|f\|_{(M)}\) will be extremal for some functional \(l(f)\) which is a linear combination of the functionals \(l_1(f),\ldots,l_m(f)\). If \(v_1(z),\ldots,v_m(z)\) are rational, then the function \(f^*(z)\) has the form

\[ f^*(z)=\|f^*\|_{(M)}\prod'\frac{z-\alpha_i}{1-\overline{\alpha_i}z} \exp\left[ \frac1{2\pi}\int_0^{2\pi} \ln\left\{ S^{-1}\left[ |A|\frac{\prod_1^{n-1}|e^{i\theta}-\alpha_i|^2} {\prod_1^n|e^{i\theta}-\beta_i|^2} \right]\right\} \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\theta \right], \tag{12} \]

where \(\beta_1,\ldots,\beta_n\) are all the poles of the functions \(v_1,\ldots,v_m\); \(\alpha_1,\ldots,\alpha_{n-1}\) are certain points in the disk \(|z|<1\); \(\prod'\) extends over some of the \(\alpha_i\); \(A\) satisfies condition (11).

Theorem 9. For any \(f(z)\in H_M^*\) there exists a sequence \(\{\varphi_n(z)\}\in H_M^*\) such that \(\{\varphi_n(z)\}\) converges uniformly inside \(|z|<1\) to \(f(z)\), and moreover

\[ \|\varphi_n\|_{(M)}=\|f\|_{(M)} \tag{13} \]

and the functions \(\varphi_n(z)\) have the form (9).

Denote by \(H_{(M)}^{*1}(a)\) the set consisting of those functions \(f(z)\in H_{(M)}^{*1}\) for which \(\|f\|_{(M)}\leqslant 1\), \(f(0)=0\), and \(f'(0)=a\ne0\). If \(|a|>M^{-1}(1/2\pi)\), then this set is empty. For \(|a|=M^{-1}(1/2\pi)\) this set consists of the single function \(f(z)=az\).

For \(|a|<M^{-1}(1/2\pi)\) there exists a disk \(|z|<r<1\) such that all functions of the class \(H_{(M)}^{*1}(a)\) are univalent in it, while for any \(R\), \(R>r\), there are functions from \(H_{(M)}^{*1}(a)\) that are not univalent in the disk \(|z|<R\). The number \(r\) will be called the radius of univalence of the class \(H_{(M)}^{*1}(a)\). A function \(f^*(z)\in H_{(M)}^{*1}(a)\) will be called extremal in the problem on the radius of univalence if it is not univalent in any disk \(|z|<R\), where \(R>r\). The notions of the radii of convexity and starlikeness of the class \(H_{(M)}^{*1}(a)\), and of extremal functions in the problems on the radii of convexity and starlikeness, are introduced analogously (cf., for bounded functions, for example, \({}^{(12)}\)).

Theorem 10. The extremal function \(f^*(z)\in H_{(M)}^{*1}(a)\) in the problem on the radius of univalence of the class \(H_{(M)}^{*1}(a)\) has the form:

\[ f^*(z)=\prod{}' \frac{z-\alpha_i}{1-\overline{\alpha_i}\,z}\, \exp\left[ \frac{1}{2\pi}\int_0^{2\pi} \ln\left\{ S^{-1}\left[ |A|\, \frac{\prod_1^3 |e^{i\theta}-\alpha_i|^2} {|e^{i\theta}-\beta_1|^2\,|e^{i\theta}-\beta_2|^2} \right] \right\} \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\theta \right]. \]

Theorem 11. The extremal function \(f^*(z)\in H_{(M)}^{*1}(a)\) in the problem on the radius of starlikeness of the class \(H_{(M)}^{*1}(a)\) has the form

\[ f^*(z)=\prod{}' \frac{z-\alpha_i}{1-\overline{\alpha_i}\,z}\, \exp\left[ \frac{1}{2\pi}\int_0^{2\pi} \ln\left\{ S^{-1}\left[ |A|\, \frac{\prod_1^3 |e^{i\theta}-\alpha_i|^2} {|e^{i\theta}-\beta|^4} \right] \right\} \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\theta \right]. \]

Theorem 12. The extremal function \(f^*(z)\in H_{(M)}^{*1}(a)\) in the problem on the radius of convexity of the class \(H_{(M)}^{*1}(a)\) has the form

\[ f^*(z)=\prod{}' \frac{z-\alpha_i}{1-\overline{\alpha_i}\,z}\, \exp\left[ \frac{1}{2\pi}\int_0^{2\pi} \ln\left\{ S^{-1}\left[ |A|\, \frac{\prod_1^4 |e^{i\theta}-\alpha_i|^2} {|e^{i\theta}-\beta|^6} \right] \right\} \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\theta \right]. \]

In these theorems \(\alpha_i\) and \(\beta_i\) are certain points inside the disk \(|z|<1\). The proof of Theorems 10–12 is based on Theorem 7 and on the methods of \({}^{(12)}\).

We note that a number of the theorems in the paper were formulated using the Luxemburg norm; analogous theorems also hold for the Orlicz norm.

In conclusion I take the opportunity to express my sincere gratitude to S. Ya. Khavinson for suggesting the topic and for consultations.

Moscow Civil Engineering Institute
named after V. V. Kuibyshev

Received
1 VIII 1961

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