Abstract
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MATHEMATICS
I. I. IBRAGIMOV and A. S. DZHAFAROV
ON SOME INEQUALITIES FOR AN ENTIRE FUNCTION OF FINITE DEGREE AND ITS DERIVATIVES*
(Presented by Academician V. I. Smirnov on 9 I 1961)
S. N. Bernstein ([1], p. 269) proved that in the class \(B_\nu\) of entire functions of degree \(\leqslant \nu\), bounded on the whole real axis, the following inequality holds:
\[ \sup_{-\infty<x<\infty} |f'(x)| \leqslant \nu \sup_{-\infty<x<\infty} |f(x)|. \tag{1} \]
Among the numerous generalizations of this inequality we cite the following (see [2], p. 154):
If \(f(z)\in B_\nu\), then for any real \(\alpha\),
\[ \sup_{-\infty<x<\infty} |f'(x)\sin\alpha+\nu f(x)\cos\alpha| \leqslant \nu \sup_{-\infty<x<\infty} |f(x)|. \tag{2} \]
Let us note that inequality (2) is sharp for every real \(\alpha\).
Denote by \(W_{\nu_1,\ldots,\nu_n}^{(p)}\) \((p\geqslant 1)\) the class of entire functions \(f(z_1,\ldots,z_n)\) of degree not exceeding \(\nu_1,\ldots,\nu_n\), for which the condition
\[ \bigl(\|f\|_{p}^{(n)}\bigr)^p = \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} |f(x_1,\ldots,x_n)|^p\,dx_1\cdots dx_n < +\infty \]
is satisfied for \(1\leqslant p<+\infty\), and
\[ \|f\|_{\infty}^{(n)} = \sup_{-\infty<x_1,\ldots,x_n<+\infty} |f(x_1,\ldots,x_n)| < +\infty \]
for \(p=+\infty\).
For an entire function \(f(z_1,\ldots,z_n)\) from the class \(W_{\nu_1,\ldots,\nu_n}^{(p)}\) \((1\leqslant p<p'\leqslant\infty)\), the inequality of S. M. Nikol’skii [3] holds:
\[ \|f\|_{p'}^{(n)} \leqslant 2^n \prod_{k=1}^{n} \nu_k^{\,1/p-1/p'} \, \|f\|_{p}^{(n)} . \tag{3} \]
Inequality (3) was sharpened in the works [4–6], and it was proved that for an entire function \(f(z_1,\ldots,z_n)\in W_{\nu_1,\ldots,\nu_n}^{(p)}\) \((1\leqslant p<p'\leqslant\infty)\),
\[ \|f\|_{p'}^{(n)} \leqslant \begin{cases} \displaystyle \prod_{k=1}^{n} \left(\frac{\nu_k}{\pi}\right)^{1/p-1/p'} \|f\|_{p}^{(n)}, & (0<p\leqslant 2), \\[1.2em] \displaystyle \prod_{k=1}^{n} \left(\frac{p\nu_k}{\pi}\right)^{1/p-1/p'} \|f\|_{p}^{(n)}, & (p>2). \end{cases} \tag{4} \]
* The results of this note were reported at the Fifth All-Union Conference on the Theory of Functions of a Complex Variable in Yerevan in September 1960.
Inequality (4) is further refined by us as follows:
Theorem 1. Let \(f(z_1,\ldots,z_n)\in W^{(p)}_{\nu_1,\ldots,\nu_n}\) \((1\leq p<p'\leq\infty)\);
\[ B_q=\left(\int_0^\infty \left|\frac{\sin t}{t}\right|^q\,dt\right)^{1/q}; \qquad B_\infty=\max_{-\infty<t<\infty}\left|\frac{\sin t}{t}\right|=1; \]
\(s\) is the least integer not smaller than \(p/2\), and the number \(q\) is chosen from the condition \(1/q+s/p=1\).
Then the inequality holds
\[ \|f\|_{p'}^{(n)} \leq \left(2^{1/q}\pi^{-1}B_q\right)^{\frac{n}{s}\left(1-\frac{p}{p'}\right)} s^{\,n\left(\frac{1}{p}-\frac{1}{p'}\right)} \prod_{k=1}^{n}\nu_k^{\frac{1}{p}-\frac{1}{p'}} \|f\|_{p}^{(n)}. \tag{5} \]
In particular, taking into account that \(B_q^q\leq B_2^2=\pi/2\) \((q\geq 2)\), from (5) we find
\[ \|f\|_{p'}^{(n)} \leq \prod_{k=1}^{n} \left(\frac{s\nu_k}{\pi}\right)^{1/p-1/p'} \|f\|_{p}^{(n)}. \tag{6} \]
Inequality (6) in the one-dimensional case was obtained independently of us by A. F. Timan (see (7), p. 248, inequality (29)).
From inequalities (2) and (5), for \(p'=\infty\) it follows immediately that, for an entire function \(f(z)\in W_\nu^{(p)}\) \((p\geq 1)\), the inequality holds
\[ \sup_{-\infty<x<\infty} |af'(x)+b\nu f(x)| \leq \nu\sqrt{a^2+b^2}\, \left(2^{1/q}\pi^{-1}B_q\right)^{1/s} s^{1/p}\nu^{1/p}\|f\|_p. \tag{7} \]
In the case \(p=2\), inequality (7) is exact only when \(a=0\).
The present note is devoted to the refinement and generalization of inequality (7).
Introduce the notation:
\[ (\alpha,\beta)= \begin{cases} \alpha-\beta, & \text{if } \alpha>\beta,\\ 0, & \text{if } \alpha\leq\beta, \end{cases} \]
\[ D_\nu[f;z;a,b;\alpha',\alpha''] = a\prod_{k=1}^{n}\nu_k^{(\alpha_k'',\alpha_k')} \frac{\partial^{\alpha_1'+\cdots+\alpha_n'}f(z)} {\partial x_1^{\alpha_1'}\cdots \partial x_n^{\alpha_n'}} + \]
\[ + b\prod_{k=1}^{n}\nu_k^{(\alpha_k',\alpha_k'')} \frac{\partial^{\alpha_1''+\cdots+\alpha_n''}f(z)} {\partial x_1^{\alpha_1''}\cdots \partial x_n^{\alpha_n''}}, \]
where \(z=(z_1,\ldots,z_n)\); \(a\) and \(b\) are complex numbers; \(\alpha_k'\) and \(\alpha_k''\) \((k=1,2,\ldots,n)\) are nonnegative integers.
Theorem 2. For an entire function \(f(z_1,\ldots,z_n)\in W^{(p)}_{\nu_1,\ldots,\nu_n}\) \((1\leq p\leq 2)\), we have
\[ |D_\nu[f;z;a,b;\alpha',\alpha'']| \leq \frac{C_q}{\pi^n} \prod_{k=1}^{n} \nu_k^{\max(\alpha_k',\alpha_k'')+1/p} \|f\|_{p}^{(n)}, \tag{8} \]
where
\[ C_q= \left\{ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \left| a\prod_{k=1}^{n} \frac{\partial^{\alpha_k'}}{\partial x_k^{\alpha_k'}} \left(\frac{\sin(x_k+i\nu_k y_k)}{x_k+i\nu_k y_k}\right) + \right. \]
\[ \left. + b\prod_{k=1}^{n} \frac{\partial^{\alpha_k''}}{\partial x_k^{\alpha_k''}} \left(\frac{\sin(x_k+i\nu_k y_k)}{x_k+i\nu_k y_k}\right) \right|^q dx_1\cdots dx_n \right\}^{1/q} \]
and the number \(q\) is determined from the condition \(1/p+1/q=1\).
It is obvious that, when \(y_k=0\) \((k=1,2,\ldots,n)\), \(p=2\), and the numbers \(a\) and \(b\) are real, if for at least one value of \(k\) the numbers \(\alpha'_k\) and \(\alpha''_k\) have different parity, inequality (8) takes the form:
\[ \left|D_{\nu}\{f;x;a,b;\alpha',\alpha''\}\right|\leq \]
\[ \leq \left[ a^2\prod_{k=1}^{n}\frac{1}{2\alpha'_k+1} + b^2\prod_{k=1}^{n}\frac{1}{2\alpha''_k+1} \right]^{-1/2} \frac{1}{\pi^{n/2}} \prod_{k=1}^{n} \nu_k^{\max(\alpha'_k,\alpha''_k)+1/2} \left\|f\right\|_2^{(n)}, \]
where \(x=(x_1,\ldots,x_n)\).
The last inequality becomes an equality for \(x_1=\cdots=x_n=0\) for the function
\[ f_0(x_1,\ldots,x_n) = a\prod_{k=1}^{n} \nu_k^{(\alpha''_k,\alpha'_k)} \left(\frac{\sin \nu_k x_k}{x_k}\right)^{(\alpha'_k)} + b\prod_{k=1}^{n} \nu_k^{(\alpha'_k,\alpha''_k)} \left(\frac{\sin \nu_k x_k}{x_k}\right)^{(\alpha''_k)}, \]
which is an entire function from the class \(W^{(2)}_{\nu_1,\ldots,\nu_n}\).
In particular, let \(b=0\), \(1\leq p\leq 2\), \(y_k=0\) \((k=1,2,\ldots,n)\), and let \(\alpha'_k=\alpha_k\) be arbitrary nonnegative numbers, while \(\alpha''_k=0\) \((k=1,2,\ldots,n)\). In this case, for an entire function \(f(z_1,\ldots,z_n)\in W^{(p)}_{\nu_1,\ldots,\nu_n}\) \((1\leq p\leq 2)\), we have
\[ \left| \frac{\partial^{\alpha_1+\cdots+\alpha_n}f(x_1,\ldots,x_n)} {\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \right| \leq \frac{A_q}{\pi^n} \prod_{k=1}^{n}\nu_k^{\alpha_k+1/p} \left\|f\right\|_p^{(n)}, \tag{9} \]
where
\[ A_q=\prod_{k=1}^{n} \left\| \left(\frac{\sin x}{x}\right)^{(\alpha_k)} \right\|_q . \]
For \(p=2\), inequality (9) becomes an equality for the function
\[ \prod_{k=1}^{n} \left(\frac{\sin kx}{kx}\right)^{(\alpha_k)} \quad \text{when } x_1=\cdots=x_n=0. \]
From inequality (9) one obtains the inequality that was previously obtained in papers \((^5,^6)\):
\[ \left| \frac{\partial^{\alpha_1+\cdots+\alpha_n}f(x_1,\ldots,x_n)} {\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \right| \leq \prod_{k=1}^{n} \nu_k^{\alpha_k} \left(\frac{\nu_k}{\pi}\right)^{1/p} (p\alpha_k+1)^{-1/p} \left\|f\right\|_p^{(n)} . \]
Theorem 3. Let \(s\) be the smallest integer not less than \(p/2\) \((p\geq 1)\), let the number \(q\) be chosen from the condition \(1/q+s/p=1\), and let \(y_k\) be arbitrary real numbers. Then, for an entire function \(f(z_1,\ldots,z_n)\in W^{(p)}_{\nu_1,\ldots,\nu_n}\) \((p\geq 1)\), the inequality
\[ \left|f(x_1+iy_1,\ldots,x_n+iy_n)\right| \leq \prod_{k=1}^{n} \left(\frac{s\nu_k}{\pi}\right)^{1/p} \left[ \frac{\operatorname{sh}(p\nu_k y_k)}{p\nu_k y_k} \right]^{1/p} \left\|f\right\|_p^{(n)} \tag{10} \]
holds.
Inequality (10) is a refinement of the corresponding inequality obtained in paper \((^8)\).
Let \(\varphi(x_1,\ldots,x_n)\geq 1\) be a fixed function, continuous
in the \(n\)-dimensional Euclidean space \(R_n\):
\[ \alpha(t_1,\ldots,t_n)= \sup_{\substack{-\infty<x_1,\ldots,x_n<\infty\\ |y_1|\le t_1,\ldots,|y_n|\le t_n}} \frac{\varphi(x_1+y_1,\ldots,x_n+y_n)} {\varphi(x_1,\ldots,x_n)} \le \]
\[ \le \sum_{k_1=0}^{m_1}\cdots\sum_{k_n=0}^{m_n} A_{k_1,\ldots,k_n}t_1^{k_1}\cdots t_n^{k_n} \equiv M(t_1,\ldots,t_n). \]
Denote by \(W_{\nu_1,\ldots,\nu_n}^{(p;\varphi)}\) the class of entire functions \(f(x_1,\ldots,x_n)\) of degree \(\nu_1,\ldots,\nu_n\), for which the following Lebesgue integral is finite:
\[ (\|f\|_{p,\varphi}^{(n)})^p = \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \left| \frac{f(x_1,\ldots,x_n)} {\varphi(x_1,\ldots,x_n)} \right|^p \,dx_1\cdots dx_n. \]
Theorem 4. If \(a,b\) are real numbers, each of the numbers \(\alpha_k'\), \(\alpha_k''\) \((k=1,2,\ldots,n)\) independently of the others assumes the values zero and one, and, moreover, for at least one value of \(k\) the numbers \(\alpha_k'\) and \(\alpha_k''\) are not equal, then
\[ \frac{\left|D_{\nu+\lambda}[f;x;a,b;\alpha',\alpha'']\right|} {\varphi(x_1,\ldots,x_n)} \le \]
\[ \le \left[ a^2\prod_{k=1}^{n}\frac{1}{2\alpha_k'+1} + b^2\prod_{k=1}^{n}\frac{1}{2\alpha_k''+1} \right]^{1/2} \times \]
\[ \times \pi^{-n/2} \prod_{k=1}^{n} (\nu_k+\lambda_k)^{\max(\alpha_k',\alpha_k'')+1/2} M\left(\frac{1}{\lambda_1},\ldots,\frac{1}{\lambda_n}\right) \|f\|_{2,n}^{(n)}, \]
where \(\lambda_k\) are arbitrary positive parameters; \(D_{\nu+\lambda}[f;x;a,b;\alpha',\alpha'']\) is obtained from the expression \(D_\nu[f;x;a,b;\alpha',\alpha'']\) by replacing \(\nu_k\) by \(\nu_k+\lambda_k\) \((k=1,2,\ldots,n)\). In the case \(\varphi(x_1,\ldots,x_n)=\prod_{k=1}^{n}(1+x_k^2)\), \(\lambda_k=\sqrt{\nu_k}\), and as \(\nu_k\to\infty\), the last inequality turns into an asymptotic equality at \(x_1=x_2=\cdots=x_n=0\) for the function \(g_0(x_1,\ldots,x_n)=\varphi(x_1,\ldots,x_n)f_0(x_1,\ldots,x_n)\), which is an entire function of degree \(\nu_1,\ldots,\nu_n\), where \(f_0(x_1,\ldots,x_n)\) is defined above.
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
5 I 1961
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