[Paper Review] Three equivalent conjectures on an estimate of integrals (in Russian)

Rustam Baladai, Б. Н. Хабибуллин|arXiv (Cornell University)|Jun 26, 2010

Meromorphic and Entire Functions3 citations

TL;DR

This paper proposes and investigates three equivalent conjectures concerning sharp estimates of improper integrals involving a parameter $\lambda > 0$, motivated by extremal problems in complex analysis. It proves the sharp estimate for $\lambda \leq 1$ and provides a non-sharp estimate for $\lambda > 1$, with equivalence established between the original conjecture and two reformulations involving integrals of increasing functions and their derivatives.

ABSTRACT

We offer a conjecture on sharp estimation of a definite improper integral depend on a parameter $λ\in (0,+\infty)$ by means of given estimate of other definite integral depend on parameters $t\in [0,+\infty)$ and $λ$. Such sharp estimate is proved for $λ\leq 1$. Besides, an estimate is obtained for $λ>1$. The last estimate is not exact seemingly. We give also two conjectures that are equivalent to the original conjecture. Sources of our conjectures are extremal problems for entire, meromorphic, and plurisubharmonic functions of several variables.

Motivation & Objective

• To establish a sharp estimate for a class of improper integrals depending on a parameter $\lambda \in (0, \infty)$, motivated by extremal problems for entire, meromorphic, and plurisubharmonic functions.
• To prove the sharpness of the estimate for $\lambda \leq 1$ using integral transforms and special functions.
• To demonstrate the equivalence of the original conjecture to two reformulations involving integrals of increasing functions and their derivatives.
• To provide a non-sharp estimate for $\lambda > 1$, suggesting it may not be optimal.
• To connect the conjecture to the classical Paley problem in several complex variables through integral inequalities.

Proposed method

• Use of integral transforms and substitution techniques to relate the original integral to beta and gamma functions.
• Application of integration by parts to derive an equivalent conjecture involving the derivative of the increasing function.
• Transformation of variables via $x^2 = x'$, $t^2 = t'$, and $\alpha = \lambda/2$ to simplify the integral expressions.
• Employment of the beta function $\mathrm{B}(a,b)$ and gamma function $\Gamma(a)$ to evaluate exact integrals for the extremal function $S_{\lambda,n}(t) = c_{\lambda,n} t^\lambda$.
• Establishment of equivalence between three conjectures: one on the original integral, one on a transformed integral of $h(t)$, and one on the integral of the derivative $q(t) = h'(t)$.
• Use of the inequality $b(x) \leq e(1+x)$ for $b(x)$ defined piecewise to bound the estimate in the $\lambda > 1$ case.

Experimental results

Research questions

• RQ1 Is the proposed upper bound for the integral $\int_0^{\infty} S(t) \frac{t^{2\lambda-1}}{(1+t^{2\lambda})^2} dt$ sharp under the condition $\int_0^1 S(tx)(1-x^2)^{n-2}x\,dx \leq t^\lambda$?
• RQ2 For which values of $\lambda$ is the estimate in the conjecture sharp, and what is the nature of the extremal function achieving equality?
• RQ3 How are the three conjectures—on $S(t)$, $h(t)$, and $q(t)$—equivalent, and what transformation links them?
• RQ4 Can the estimate for $\lambda > 1$ be improved, or is it inherently non-sharp?
• RQ5 What is the connection between the integral inequality and extremal problems for entire, meromorphic, and plurisubharmonic functions in $\mathbb{C}^n$?

Key findings

• The sharp estimate $\int_0^{\infty} S(t) \frac{t^{2\lambda-1}}{(1+t^{2\lambda})^2} dt \leq \frac{\pi(n-1)}{2\lambda} \prod_{k=1}^{n-1} \left(1 + \frac{\lambda}{2k}\right)$ is proven for $\lambda \leq 1$.
• Equality in the estimate is achieved for the extremal function $S_{\lambda,n}(t) = 2(n-1) \prod_{k=1}^{n-1} \left(1 + \frac{\lambda}{2k}\right) t^\lambda$.
• For $\lambda > 1$, a non-sharp estimate is obtained: $\int_0^{\infty} S(t) \frac{t^{2\lambda-1}}{(1+t^{2\lambda})^2} dt \leq \frac{\pi}{2} \cdot \frac{n-1}{\lambda} \left(1 + \frac{1}{2} \cdot \frac{\lambda}{n-1}\right)^{n-1} \left(1 + 2 \cdot \frac{n-1}{\lambda}\right)^{\lambda/2}$.
• The conjecture is equivalent to a reformulation involving the function $h(t)$, where $\int_0^1 \frac{h(tx)}{x} (1-x)^{n-1} dx \leq t^\alpha$ implies $\int_0^{\infty} \frac{h(t)}{t} \frac{dt}{1+t^{2\alpha}} \leq \frac{\pi}{2} \prod_{k=1}^{n-1} \left(1 + \frac{\alpha}{k}\right)$ with $\alpha = \lambda/2$.
• A third equivalent conjecture involves the derivative $q(t) = h'(t)$, where $\int_0^1 \left( \int_x^1 (1-y)^{n-1} \frac{dy}{y} \right) q(tx) dx \leq t^{\alpha-1}$ implies $\int_0^{\infty} q(t) \log\left(1 + \frac{1}{t^{2\alpha}}\right) dt \leq \frac{\pi}{\mathrm{B}(\alpha,n)}$.
• The estimate for $\lambda > 1$ is not sharp, as shown by the use of the function $b(x)$ satisfying $b(x) \leq e(1+x)$, which leads to a looser bound than the sharp one for $\lambda \leq 1$.