MATHEMATICS

I. I. IBRAGIMOV, D. I. MAMEDKHANOV

RELATION BETWEEN WEIGHTED NORMS OF AN ENTIRE FUNCTION OF FINITE DEGREE ON LINES PARALLEL TO THE REAL AXIS

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

Let \(f(z)\) be an entire function of finite degree \(\sigma\), and let \(\varphi(x) \geqslant 1\) be a continuous function on the entire real axis. Along with the already considered classes \(B_{\sigma}^{(\varphi)}\) and \(W_{\sigma}^{(p,\varphi)}\) of entire functions of finite degree (see \((^1)\), pp. 37–38), we introduce into consideration also the class \(Б_{\sigma}^{(p,\varphi)}\) of entire functions \(f(z)\) of finite degree \(\sigma\) satisfying the conditions:

1) \(|f(x+iy)| \leqslant |f(x-iy)|\);

2)
\[ \|f\|_{p,\varphi} = \left( \int_{-\infty}^{\infty} \left|\frac{f(x)}{\varphi(x)}\right|^p dx \right)^{1/p} <+\infty \quad (p \geqslant 1). \]

In the case \(\varphi \equiv 1\), the classes \(B_{\sigma}^{(1)}\), \(Б_{\sigma}^{(p,1)}\), and \(W_{\sigma}^{(p,1)}\) are denoted respectively by \(B_{\sigma}\), \(Б_{\sigma}^{(p)}\), and \(W_{\sigma}^{(p)}\).

In the present note, first, for entire functions \(f(z)\) from the class \(W_{\sigma}^{(p,\varphi)}\) an estimate is given for the norm

\[ \left\| \frac{1}{\varphi(x)} \left[ f(x+iy)e^{-i\omega}+f(x-iy)e^{i\omega} \right] \right\|_{p} = \left\| f(x+iy)e^{-i\omega}+f(x-iy)e^{i\omega} \right\|_{p,\varphi}, \]

where \(\omega\) is a real parameter, in terms of the norm \(\|f\|_{p,\varphi}\), which is of auxiliary character. In the special case \(\varphi \equiv 1\), this problem was considered by P. Boas \((^2)\).

Secondly, for a function \(f(z)\in Б_{\sigma}^{(p,\varphi)}\), a relation is found between the norms \(\|f(x+iy)\|_{p,\varphi}\) and \(\|f(x)\|_{p,\varphi}\) in the form of an inequality. This problem was also considered in the special case, when \(\varphi \equiv 1\), by P. Boas and K. Rahman \((^3)\).

Further, for a function \(f(z)\in Б_{\sigma}^{(p,\varphi)}\), an inequality of S. M. Nikol’skii type is established; namely, an inequality is found between the norms \(\|f(x+iy)\|_{p',\varphi}\) and \(\|f(x)\|_{p,\varphi}\), where \(1 \leqslant p < p' \leqslant \infty\).

For the solution of the indicated problems, the following conditions are imposed on the weight function \(\varphi(x)\):

\[ \alpha_{\varphi}(t) = \sup_{\substack{-\infty<x<\infty\\ |y|\leqslant t}} \frac{\varphi(x+y)}{\varphi(x)} \leqslant P_m(t) = \sum_{k=0}^{m} A_k t^k, \tag{1} \]

where \(A_k \geqslant 0\) \((k=0,1,\ldots,m)\) are the coefficients of the polynomial \(P_m(t)\). In addition, the notation

\[ M=\sum_{k=0}^{m} A_k \]

is used.

1. Let \(f(z)\in W_{\sigma}^{(p,\varphi)}\), and consider the auxiliary function\(^*\)

\[ g(z)=f(z)\left[\frac{\sin(z-t)}{z-t}\right]^m, \tag{2} \]

\(^*\) In the papers \((^1,^5)\), in the expression \(g(z)\), instead of \(z-t\) one took \(\lambda(z-t)\), where \(\lambda>0\) is a real number. We have observed that, without diminishing the generality of the arguments carried out in solving similar problems, one may always assume \(\lambda=1\).

where \(m>0\) is an integer, and \(t\) is a real parameter. It turns out that, for the function \(g(z)\), the inequalities

\[ \|g\|_{C}=\sup_{-\infty<x<\infty}|g(x)|\leq M\varphi(t)\|f\|_{C,\varphi}, \tag{3} \]

\[ \|g\|_{p}=\|g(x)\|_{p}\leq M\varphi(t)\|f\|_{p,\varphi} \tag{4} \]

hold for every fixed \(t\), where

\[ \|f\|_{C,\varphi}=\sup_{-\infty<x<\infty}\left|\frac{f(x)}{\varphi(x)}\right|. \]

It follows from this that P. Boas’ interpolation formula is applicable to the function \(g(z)\) (see \((^{1})\), p. 137):

\[ g(x+iy)e^{-i\omega}+g(x-iy)e^{i\omega} = 2\sum_{-\infty}^{\infty}(-1)^n C_n g\left(x-s+\frac{n\pi}{\sigma+m}\right), \tag{5} \]

where \(\omega\) is a real parameter,

\[ C_n=\frac{\mu y\,\operatorname{Im}\{e^{-i\omega}\sin(s+i\mu y)\}}{(n\pi-s\mu)^2+\mu^2y^2} \]

and \(s\) is determined from the equality

\[ s\mu=\arg\{\cos(\omega+i\mu y)\},\qquad \mu=\sigma+m. \]

At the same time one can show that

\[ |g(u+\mu_n)|\leq \]

\[ \leq \varphi(t)\left|\frac{f(u+\mu_n)}{\varphi(u+\mu_n)}\right| \alpha_\varphi(|u+\mu_n-t|) \left[\frac{\sin(u+\mu_n-t)}{u+\mu_n-t}\right]^m \leq M\varphi(t)\left|\frac{f(u+\mu_n)}{\varphi(u+\mu_n)}\right|, \]

where \(\mu_n=n\pi/(\sigma+m)\), \(u=x-s\).

Owing to this, from (5) we obtain

\[ \left| \frac{f(x+iy)e^{-i\omega}+f(x-iy)e^{i\omega}}{\varphi(x)} \right| \cdot \left|\frac{\sin iy}{iy}\right|^m \leq 2M\sum_{-\infty}^{\infty}|C_n| \left| \frac{f(x-s+\mu_n)}{\varphi(x-s+\mu_n)} \right|. \tag{6} \]

Thus, taking into account that

\[ \sum_{-\infty}^{\infty}|C_n| = \left[\operatorname{ch}^2(\sigma+m)y-\sin^2\omega\right]^{1/2}, \]

the assertion follows from inequality (6):

Theorem 1. For an entire function \(f(z)\in W_\sigma^{(p,\varphi)}\), the inequality

\[ \left\|f(x+iy)e^{-i\omega}+f(x-iy)e^{i\omega}\right\|_{p,\varphi} \leq \]

\[ \leq 2M\left(\frac{y}{\operatorname{sh}y}\right)^m \left[\operatorname{ch}^2(\sigma+m)y-\sin^2\omega\right]^{1/2} \|f\|_{p,\varphi}. \tag{7} \]

2. Let us note that inequality (7) remains valid also for a function \(f(z)\) from the class \(B_\sigma^{(p,\varphi)}\); it can be written in the form

\[ \int_{-\infty}^{\infty} \left|\frac{f(x+iy)}{\varphi(x)}\right|^p \left|1+\lambda(x,y)e^{2i\omega}\right|^p\,dx \leq \]

\[ \leq 2^p M^p \left(\frac{y}{\operatorname{sh}y}\right)^{mp} \left[\operatorname{ch}^2(\sigma+m)y-\sin^3\omega\right]^{p/2} \|f\|_{p,\varphi}, \]

where

\[ \lambda(x,y)=\frac{f(x-iy)}{f(x+iy)}=\rho(x,y)e^{i\theta},\qquad 0<\rho(x,y)\leq 1. \]

Integrating both sides of the last inequality with respect to \(\omega\) from \(0\) to \(2\pi\) and observing that

\[ \int_{0}^{2\pi}\left|1+\rho(x,y)e^{i(\theta+2\omega)}\right|^p\,d\omega \geq 2^{p+1}B\left(\frac{1}{2}p+\frac{1}{2},\frac{1}{2}\right), \]

after slight transformations we arrive at the following conclusion:

Theorem 2. For an entire function \(f(z)\in B_\sigma^{(p,\varphi)}\) the inequality

\[ \|f(x+iy)\|_{p,\varphi} \leq M D_p[(\sigma+m)y]\operatorname{ch}(\sigma+m)y \left(\frac{y}{\operatorname{sh}y}\right)^m \|f\|_{p,\varphi}, \tag{8} \]

holds, where

\[ D_p(u)= \left\{ \frac{1}{2B\left(\frac{1}{2}p+\frac{1}{2},\frac{1}{2}\right)} \int_{0}^{2\pi}\left(1-\sin^2\omega\,\operatorname{sech}^2u\right)^{p/2}\,d\omega \right\}^{1/p}, \tag{9} \]

and \(B(\alpha,\gamma)\) is Euler’s beta function.

1. For an entire function \(f(z)\in B_\sigma^{(p,\varphi)}\), consider again the auxiliary function \(g(z)\) defined by equality (2), and note that it satisfies the condition
   \(|g(x+iy)|\leq |g(x-iy)|\); as inequality (4) shows, \(g(z)\) belongs to the class \(W_{\sigma+m}^{(p)}\). In paper (4) it is proved that for \(g(z)\in W_{\sigma+m}^{(p)}\) the inequality

\[ \sup_{-\infty<x<\infty}|g(x+iy)| \leq [\omega_p(\sigma+m,y)]^{1/p}\|g\|^p, \tag{10} \]

holds, where

\[ \omega_p(\mu,y)= \begin{cases} \dfrac{\operatorname{sh}p\mu y}{\pi p y}, & \text{for } 1\leq p\leq 2,\\[6pt] \dfrac{\operatorname{sh}p\mu y}{\pi y}, & \text{for } p>2. \end{cases} \]

We note that for entire functions \(f(z)\) from the class \(B_\sigma^{(p)}\), the expression \(\omega_p(\mu,y)\) was refined by D. I. Mamedkhanov; namely, it was found that

\[ \omega_p(\mu,y)=\frac{s\mu}{\pi}\,[D_{p/s}(s\mu y)\operatorname{ch}s\mu y]^{p/s}, \]

where \(\mu=\sigma+m\) and \(D_p(u)\) is defined by equality (9).

From inequality (10), taking into account inequality (4) and then putting \(t=x\), we obtain:

\[ \left|\frac{f(x+iy)}{\varphi(x)}\right| \leq M\left(\frac{y}{\operatorname{sh}y}\right)^m \left(\frac{s\mu}{\pi}\right)^{1/p} [D_{p/s}(s\mu y)\operatorname{ch}s\mu y]^{1/s}. \tag{11} \]

Thus, from the inequality

\[ \|f(x+iy)\|_{p',\varphi} \leq \left\{ \sup_{-\infty<x<\infty} \left|\frac{f(x+iy)}{\varphi(x)}\right| \right\}^{(p'-p)/p'} \|f(x+iy)\|_{p,\varphi}^{p/p'} \]

for \(1\leq p<p'\leq\infty\), by virtue of inequalities (8) and (11), we arrive at the following conclusion:

Theorem 3. For an entire function \(f(z)\in B_\sigma^{(p,\varphi)}\), with \(1\leq p<p'\leq\infty\), the inequality

\[ \|f(x+iy)\|_{p',\varphi} \leq M\left(\frac{s\mu}{\pi}\right)^{1/p-1/p'} \left(\frac{y}{\operatorname{sh}y}\right)^m [D_{p/s}(s\mu y)\operatorname{ch}s\mu y]^{(p'-p)/sp'} [D_p(\mu y)\operatorname{ch}\mu y]^{p/p'} \|f\|_{p,\varphi}, \]

holds, where \(\mu=\sigma+m\) and \(D_p(u)\) is defined by equality (9).

Finally, let us denote by \(B_{\sigma_1,\ldots,\sigma_n}^{(p,\varphi)}\) the class of entire functions \(f(z_1,\ldots,z_n)\) of finite degree \(\sigma_1,\ldots,\sigma_n\) satisfying the conditions:

\(1^\circ.\)
\[ |f(x_1+iy_1,\ldots,x_n+iy_n)|\leq |f(x_1-iy_1,\ldots,x_n-iy_n)|, \]

\(2^\circ.\)
\[ \|f\|_{p,\varphi}^{(n)} = \left( \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \left| \frac{f(x_1,\ldots,x_n)}{\varphi(x_1,\ldots,x_n)} \right|^p \,dx_1\cdots dx_n \right)^{1/p} <+\infty . \]

Here it is assumed that

\[ a_k(t)= \sup_{\substack{-\infty<x_1,\ldots,x_n<\infty\\ |y|\leq t}} \frac{\varphi(x_1,\ldots,x_{k-1},x_k+y,x_{k+1},\ldots,x_n)} {\varphi(x_1,x_2,\ldots,x_n)} \leq \sum_{j=0}^{m} A_j t^j \]

\[ (k=1,2,\ldots,n). \]

Theorem 4. For an entire function \(f(z_1,\ldots,z_n)\) from the class \(B_{\sigma_1,\ldots,\sigma_n}^{(p,\varphi)}\), the following inequalities hold:

\[ 1^\circ.\quad \|f(x_1+iy_1,\ldots,x_n+iy_n)\|_{p,\varphi}^{(n)} \leq M^n\prod_{k=1}^{n}\Omega_k(y_k)\,\|f\|_{p,\varphi}^{(n)}. \]

\[ 2^\circ.\quad \|f(x_1+iy_1,\ldots,x_n+iy_n)\|_{p',\varphi}^{(n)} \leq M^n\prod_{k=1}^{n}\mathfrak{M}_k(y_k)\,\|f\|_{p,\varphi}^{(n)} \]

for \(1\leq p<p'\leq\infty\), where \(\mu_k=\sigma_k+m\),

\[ \Omega_k(y_k)=D_p(\mu_k y_k)\operatorname{ch}\mu_k y_k \left(\frac{y_k}{\operatorname{sh} y_k}\right)^m, \]

\[ \mathfrak{M}_k(y_k)= \]

\[ = \left(\frac{s\mu_k}{\pi}\right)^{1/p-1/p'} \left(\frac{y_k}{\operatorname{sh} y_k}\right)^m \left[D_{p/s}(s\mu_k y_k)\operatorname{ch}s\mu_k y_k\right]^{(p'-p)/sp'} \left[D_p(\mu_k y_k)\operatorname{ch}\mu_k y_k\right]^{p/p'}, \]

is determined by equality (9).

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
17 II 1964

REFERENCES

¹ I. I. Ibragimov, Extremal properties of entire functions of finite degree, Baku, 1962.
² R. P. Boas, Math. Scand., 4 (1956).
³ R. P. Boas, K. Rahman, DAN, 147, No. 1 (1962).
⁴ I. I. Ibragimov, Izv. AN SSSR, ser. matem., 24, No. 4 (1960).
⁵ I. I. Ibragimov, A. S. Dzhafarov, Izv. AN AzerbSSR, ser. phys.-math. sciences, No. 5 (1962).