Zygmund's inequality for entire functions of exponential type (Q2654444)

Let \(f\in L^p(\mathbb R)\) \((p\geq 1)\) be an entire function of exponential type \(\sigma\) such that \(\limsup\limits_{r\to\infty}\frac{\ln|f(re^{i\frac{\pi}2})|}{r}\leq 0\). The author proves that \[ \int_{-\infty}^\infty|f^\prime(x)|^pdx\leq c_p\sigma^p\int_{-\infty}^\infty|\Re f(x)|^pdx, \] where the constant \(c_p=\sqrt{\pi}\frac{\Gamma(1+\frac p2)}{\Gamma\left(\frac{1+p}2\right)}\) is the best possible. Here \(\Gamma(\cdot)\) is Euler's Gamma function. This extends the well-known \textit{A. Zygmunt} inequality [J. Math. Phys. 21, 117--123 (1942; Zbl 0060.14801)] concerning \(L^p\) estimates of derivatives of polynomials.