Section 7

The Order of Growth of Certain Sequences of Polynomials

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Problem 3.263

Let h(φ)h(\varphi ) denote the support function of the infinite convex domain considered in problem 114. The sequence f1(z),f2(z),…,fn(z),…f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots

fn(z)=z(z−1)(z−2)⋯(z−n+1)(z−n)n!e−rh(φ),n=1,2,3,…, f_{n}(z) = \frac{\sqrt{z}(z-1)(z-2)\cdots (z-n+1)(z-n)}{n!}e^{-rh(\varphi )}, \quad n=1,2,3,\ldots ,

where z=reiφz=re^{i\varphi }, is bounded in the entire half-plane Re(z)≥0\mathrm{Re}\left(z\right)\geq 0 [problems 12, and Part II, problem 220]; h(φ)h(\varphi ) is the smallest function of the angle φ\varphi that keeps the sequence bounded.

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Problem 3.264

Let h(φ)h(\varphi ) denote the support function of the convex domain considered in problem 115. The sequence f1(z),f2(z),…,fn(z),…f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots

fn(z)=z(1−z212)(1−z222)⋯(1−z2n2)e−rh(φ),n=1,2,3,…, f_{n}(z) = z\left(1-\frac{z^{2}}{1^{2}}\right)\left(1-\frac{z^{2}}{2^{2}}\right)\cdots \left(1-\frac{z^{2}}{n^{2}}\right)e^{-rh(\varphi )}, \quad n=1,2,3,\ldots ,

where z=reiφz=re^{i\varphi }, is bounded in the entire zz-plane [problems 13, and Part II, problem 221]; h(φ)h(\varphi ) is the smallest function of the angle φ\varphi that keeps the sequence bounded.

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Problem 3.265

Let h(φ)h(\varphi ) denote the support function of the convex domain considered in problem 116. Then

∣1+zn∣ne−rh(φ)≤1,n=1,2,3,…, \left|1+\frac{z}{n}\right|^{n}e^{-rh(\varphi )} \leq 1, \quad n=1,2,3,\ldots ,

z=reiφz=re^{i\varphi }, in the entire plane.

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