MATHEMATICS

D. I. MAMEDKHANOV

INEQUALITIES FOR POSITIVE ENTIRE FUNCTIONS IN A GENERALIZED LEBESGUE SPACE

(Presented by Academician I. M. Vinogradov, 19 II 1964)

Let \(\mathscr L_{\mathbf p}^{(n)}(E_n)\), where \(\mathbf p=(p_1,\ldots,p_n)\) and \(1\le p_k\le \infty\) \((k=1,2,\ldots,n)\), denote the class of functions \(f(x_1,\ldots,x_n)\), measurable in the \(n\)-dimensional Euclidean space \(E_n\), satisfying the condition

\[ \|f\|_{\mathbf p}^{(n)} = \left\{ \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} \cdots \left( \int_{-\infty}^{\infty} |f(x_1,\ldots,x_n)|^{p_1}\,dx_1 \right)^{p_2/p_1} dx_2 \cdots \right]^{p_n/p_{n-1}} dx_n \right\}^{1/p_n} <\infty; \tag{1} \]

this class is called the generalized Lebesgue space (see \((^3)\)).

Denote by \(\widetilde W_{\vec\sigma}^{(\mathbf p)}\) \([\mathbf p=(p_1,\ldots,p_n);\ \vec\sigma=(\sigma_1,\ldots,\sigma_n)]\) the class of entire functions \(f(z_1,\ldots,z_n)\) of finite degree \(\vec\sigma\), positive for real values of the arguments, and belonging to the generalized Lebesgue space \(\mathscr L_{\mathbf p}^{(n)}(E_n)\).

In the present note, for functions of the class \(\widetilde W_{\vec\sigma}^{(\mathbf p)}\), sharp inequalities of the type of S. M. Nikolskii’s inequalities are obtained, which generalize and refine certain known results. In addition, a relation is established between different norms of entire functions of the class \(\widetilde W_{\vec\sigma}^{(\mathbf p)}\) on lines parallel to the real axis.

Theorem 1. If \(f(z_1,\ldots,z_n)\in \widetilde W_{\vec\sigma}^{(\mathbf p)}\), then for \(p_1\ge p_2\ge\cdots\ge p_n\) and \(1\le p_k<p'_k\le\infty\) \((k=1,2,\ldots,n)\), for the function *

\[ \psi(x_n) = \left\{ \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} \cdots \left( \int_{-\infty}^{\infty} |f(x_1,\ldots,x_n)|^{p_1}\,dx_1 \right)^{p_2/p_1} \cdots \right]^{p_{n-1}/p_{n-2}} dx_{n-1} \right\}^{1/p_{n-1}} \]

we have

\[ \|\psi\|_{p'_n} \le \left(\frac{\sigma_n s_n}{2\pi}\right)^{1/p_{n-1}-1/p'_n} \|\psi\|_{p_n}, \tag{2} \]

where \(s_n=\left|[-p_n/2]\right|\) is the least integer not less than \(p_n/2\), and

\[ \|\psi\|_q = \left( \int_{-\infty}^{\infty} |\psi(x_n)|^q\,dx_n \right)^{1/q}. \]

Theorem 2. If \(f(z_1,\ldots,z_n)\in \widetilde W_{\vec\sigma}^{(\mathbf p)}\), then for \(p_1\ge p_2\ge\cdots\ge p_n\) and \(1\le p_k<p'_k\le\infty\) \((k=1,2,\ldots,n)\) we have

\[ \|f\|_{\mathbf p'}^{(n)} \le \prod_{k=1}^{n} \left(\frac{\sigma_k s_k}{2\pi}\right)^{1/p_k-1/p'_k} \|f\|_{\mathbf p}^{(n)}, \tag{3} \]

where \(s_k=\left|[-p_k/2]\right|\) \((k=1,\ldots,n)\) is the least integer not less than \(p_k/2\), and \(\|f\|_{\mathbf p}^{(n)}\) is defined by equality (1).

* One can give an example in which the function \(\psi(x_n)\) is not an analytic function, but \(f(z_1,\ldots,z_n)\) belongs to the class \(\widetilde W_{\vec\sigma}^{(\mathbf p)}\).

We note that inequality (3) is sharp*. In the case \(p'=\infty\) and \(p=1\), equality is attained for the function

\[ f_0(x_1,\ldots,x_n)=\prod_{i=1}^n\left(\frac{\sin \sigma_i x_i/2}{x_i}\right)^2 . \]

Let us note that inequality (3) is a refinement of the corresponding inequality of I. I. Ibragimov \((^3)\) for entire functions from the class \(\widetilde W_{\sigma}^{(p)}\).

Denote by \(B_{\sigma}^{(p)}\) the class of functions from the class \(W_{\sigma}^{(p)}\) satisfying the condition

\[ |f(x+iy)|\leq |f(x-iy)|\qquad (y\geq 0). \tag{4} \]

This class is a generalization of the class of functions that are real on the real axis. For this class of entire functions we give the following theorems**.

Theorem 3. For an entire function \(f(z)\) from the class \(B_{\sigma}^{(p)}\), for any \(1\leq p\leq p'\leq \infty\), the inequality

\[ \|f(x+iy)\|_{p',x}\leq \left(\frac{s\sigma}{\pi}\right)^{1/p-1/p'} \bigl[Q_p(\sigma y)\operatorname{ch}\sigma y\bigr]^{p/p'} \bigl[Q_{p/s}(s\sigma y)\operatorname{ch}s\sigma y\bigr]^{(p'-p)/sp'} \times \]

\[ \times \|f(x)\|_{p,x}, \tag{5} \]

holds, where \(s=[-p/2]\) is the smallest integer not less than \(p/2\), and

\[ Q_r(t)= \frac{\displaystyle\int_0^{2\pi}(1-\sin^2\omega\,\operatorname{sh}^2 t)^{r/2}\,d\omega} {2B(1/2r+1/2,\,1/2)}. \tag{6} \]

Theorem 4. For a positive entire function \(f(z)\) from the class \(B_{\sigma}^{(p)*}\), for any \(1\leq p<p'\leq \infty\), the inequality

\[ \|f(x+iy)\|_{p',x}\leq \left(\frac{s\sigma}{2\pi}\right)^{1/p-1/p'} Q_p(\sigma y)\operatorname{ch}\sigma y\,\|f(x)\|_{p,x}, \tag{7} \]

holds, where \(s\) and \(Q_p\) are the same as in Theorem 3.

Inequality (5) is sharp for periodic (and, consequently, trigonometric-sum) functions, if the norms are taken over \((0,2\pi)\) instead of \((-\infty,\infty)\); in this case equality is attained for the function

\[ f(z)=\cos\sigma z \qquad \text{when } p'=p . \]

For entire functions of several variables \(f(z_1,\ldots,z_n)\) from the classes \(W_{\vec\sigma}^{(\mathbf p)}\) and \(\widetilde W_{\vec\sigma}^{(p)}\) we give the following theorems.

Theorem 5. If \(p'_1,\ldots,p'_n\) are distinct numbers not less than one, \(p_1\geq p_2\geq\cdots\geq p_n\), and \(1\leq p_k\leq p'_k\leq \infty\) \((k=1,2,\ldots,n)\), then for an entire function from the class \(W_{\vec\sigma}^{(\mathbf p)}\) the inequality

\[ \|f(x_1+iy_1,\ldots,x_n+iy_n)\|_{\mathbf p'} \leq \prod_{k=1}^n \left(\frac{s_k\sigma_k}{\pi}\right)^{1/p_k-1/p'_k} \exp\left(\sum_{k=1}^n \sigma_k |y_k|\right) \|f(x_1,\ldots,x_n)\|_{\mathbf p} \tag{8} \]

holds.

* Inequality (3) in the case \(p_1=p_2=\cdots=p_n=p\), \(p'_1=\cdots=p'_n=p'\), and \(n=1\) was obtained in \((^1,^5)\). Later a similar inequality was found by P. Boas \((^2)\) in the two-dimensional case with non-sharp constants in the case \(p_1=p_2=1\) and \(p'_1=p'_2=\infty\). This problem in the class of trigonometric polynomials was solved by N. Sabziev \((^4)\).

** These results were reported at the VII All-Union Conference on the Theory of Functions of a Complex Variable in September 1963 in Rostov-on-Don.

This result is obtained directly from \((^3,\,^5)\).

Theorem 6. Under the hypotheses of Theorem 5, for an entire function of the class \(\widetilde W_{\vec\sigma}^{(\mathbf p)}\) we have

\[ \|f(x_1+iy_1,\ldots,x_n+iy_n)\|_{\mathbf p'} \le \prod_{k=1}^{n}\left(\frac{s_k\sigma_k}{2\pi}\right)^{1/p_k-1/p'_k} \exp\left(\sum_{k=1}^{n}\sigma_k|y_k|\right) \|f(x_1,\ldots,x_n)\|_{\mathbf p}. \tag{9} \]

The collection of all entire functions of several variables \(f(z_1,\ldots,z_n)\) from the class \(W_{\vec\sigma}^{(\mathbf p)}\) satisfying the condition

\[ |f(x_1+iy_1,\ldots,x_n+iy_n)| \le |f(x_1-iy_1,\ldots,x_n-iy_n)| \tag{10} \]

where \(y_1,\ldots,y_n \ge 0\), will be denoted by \(B_{\vec\sigma}^{(\mathbf p)}\), and all positive entire functions from \(B_{\vec\sigma}^{(\mathbf p)}\) by \(\widetilde B_{\vec\sigma}^{(\mathbf p)}\). For these classes of entire functions one can sharpen inequalities (8) and (9).

Theorem 7. Under the hypotheses of Theorem 5, for an entire function of the class \(B_{\vec\sigma}^{(\mathbf p)}\) the inequality

\[ \|f(x_1+iy_1,\ldots,x_n+iy_n)\|_{\mathbf p'} \le \prod_{k=1}^{n} \left(\frac{s_k\sigma_k}{\pi}\right)^{1/p_k-1/p'_k} \left[Q_{p_k}(\sigma_k,y_k)\operatorname{ch}\sigma_ky_k\right]^{p_k/p'_k} \times \]

\[ \times \left[Q_{p_k/s_k}(s_k,\sigma_k,y_k)\operatorname{ch}(s_k\sigma_ky_k)\right]^{(p'_k-p_k)/s_kp'_k} \|f(x_1,\ldots,x_n)\|_{\mathbf p}, \tag{11} \]

where \(s_k\) and \(Q_{p_k}\) are the same as in Theorem 3.

Theorem 8. Under the hypotheses and notation of the preceding theorem, for an entire function of the class \(\widetilde B_{\vec\sigma}^{(\mathbf p)}\) we have

\[ \|f(x_1+iy_1,\ldots,x_n+iy_n)\|_{\mathbf p'} \le \prod_{k=1}^{n} \left(\frac{s_k\sigma_k}{2\pi}\right)^{1/p_k-1/p'_k} \left[Q_{p_k}(\sigma_ky_k)\operatorname{ch}\sigma_ky_k\right]^{p_k/p'_k} \times \]

\[ \times \left[Q_{p_k/s_k}(s_k,\sigma_k,y_k)\operatorname{ch}s_k\sigma_ky_k\right]^{(p'_k-p_k)/p'_ks_k} \|f(x_1,\ldots,x_n)\|_{\mathbf p}. \tag{12} \]

Let us also note that inequalities (5) and (7) remain valid for the functions

\[ \psi(x_n,y_n)= \left\{ \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} \cdots \left( \int_{-\infty}^{\infty} |f(x_1,\ldots,x_{n-1},z_n)|^{p_1}\,dx_1 \right)^{p_2/p_1} \cdots \right]^{p_{n-1}/p_{n-2}} dx_{n-1} \right\}^{1/p_{n-1}}, \]

if \(f(z_1,\ldots,z_n)\) is taken respectively from \(B_{\vec\sigma}^{(\mathbf p)}\) or \(\widetilde B_{\vec\sigma}^{(\mathbf p)}\).

Received
17 II 1964

CITED LITERATURE

\(^1\) I. I. Ibragimov, Extremal Properties of Entire Functions, Baku, 1962.
\(^2\) R. Boas, Proc. Am. Math. Soc., 13, No. 4 (1962).
\(^3\) I. I. Ibragimov, DAN, 152, No. 5 (1963).
\(^4\) N. Sabziev, Abstracts of Reports, All-Union Conference on the Constructive Theory of Functions, Baku, 1962.
\(^5\) I. I. Ibragimov, A. S. Jafarov, Izv. AN AzerbSSR, series of physical and mathematical sciences, No. 5 (1962).