Weighted Product Inequalities for the Sine Function: A Gamma-Function Approach and Sharp Comparisons

1. Problem Statement

The paper addresses the classical problem of establishing inequalities for weighted products of sine functions. Specifically, it focuses on the inequality derived from the log-concavity of the sine function on $(0, \pi)$:
$$\prod_{i=1}^n \sin^{\alpha_i}(\theta_i) \leq \sin^\alpha \left( \sum_{i=1}^n \frac{\alpha_i}{\alpha} \theta_i \right)$$
where $\alpha_i > 0$ are weights and $\alpha = \sum \alpha_i$.

While this inequality is well-known, the authors identify a gap in the literature: for a fixed product on the left-hand side (LHS), different choices of the auxiliary parameters (weights and angles) used to derive the right-hand side (RHS) can yield distinct upper bounds. The central problem is to determine which of these competing upper bounds is sharper (i.e., smaller) and to establish the exact algebraic conditions on the parameters under which one bound dominates the other.

2. Methodology

The authors employ a novel approach that avoids direct reliance on the log-concavity of the sine function. Instead, they utilize the properties of the Gamma function ($\Gamma$) and the Beta function ($B$).

• Gamma-Function Foundation: They leverage the log-convexity of the Gamma function on $(0, \infty)$, a property characterized by the Bohr–Mollerup theorem.
• Euler's Reflection Formula: They utilize the identity $B(1-x, x) = \Gamma(1-x)\Gamma(x) = \frac{\pi}{\sin(\pi x)}$. This allows them to transform inequalities involving products of Gamma values into inequalities involving products of sine values.
• Generalized Hölder's Inequality: The core of their proof relies on applying the log-convexity of $\Gamma$ (via Lemma 2.1) to derive an inequality for products of Gamma functions, which is then converted into the desired sine inequality.
• Parameterization Strategy: To generate competing bounds for the same LHS, the authors introduce two distinct sets of auxiliary arguments ($a_i$) and weights ($p_i$) into the general framework. This results in two different weighted mean angles on the RHS.
• Comparison via Lemma 4.1: To compare the two resulting bounds, $S$ and $T$, the authors reduce the problem to comparing the arguments of the sine functions. They prove a key lemma stating that for $\lambda > 0$ and $\sigma, \tau \in (0, \lambda)$:
  $$\sin\left(\frac{\pi\sigma}{\lambda}\right) \geq \sin\left(\frac{\pi\tau}{\lambda}\right) \iff (\sigma - \tau)(\sigma + \tau - \lambda) \leq 0$$
  This algebraic criterion allows for precise determination of the sharper bound based on the sign of specific polynomial expressions in the parameters.

3. Key Contributions and Results

A. Alternative Proof of Classical Inequality

The paper provides a new proof of the classical weighted sine product inequality (Inequality 3 in the text) by deriving it from the log-convexity of the Gamma function and Euler's reflection formula, rather than the direct log-concavity of sine.

B. Generation of Competing Bounds

The authors construct two distinct upper bounds for the same cyclic product of sines:

1. General $n$-angle case: Two bounds, denoted $S$ and $T$, are derived with arguments $\sigma$ and $\tau$ respectively.
2. Even $2n$-angle case: Two bounds, $P$ and $Q$, are derived for paired products.

C. Sharp Comparison Criteria

The paper provides exact algebraic conditions determining when one bound is sharper than the other:

• General $n$-angle Case (Theorem 4.1):
  For parameters $\lambda_1, \dots, \lambda_n$, let $\delta_n = \sum_{i=1}^{n-1} \lambda_{i+1}(1-2\lambda_i)$.
  The bound $S$ is sharper than or equal to $T$ ($S \geq T$) if and only if:
  $$\delta_n (2\lambda_n - 1) \leq 0$$
  Equality holds if $\delta_n = 0$ or $\lambda_n = 1/2$.

• General $2n$-angle Case (Theorem 4.2):
  For parameters $\lambda_1, \dots, \lambda_{2n}$, define $\Delta_n = \sum \lambda_{2i-1}(2\lambda_{2i}-1)$ and $\Sigma_n = \sum (1-\lambda_{2i})(2\lambda_{2i-1}-1)$.
  The bound $P$ is sharper than or equal to $Q$ ($P \geq Q$) if and only if:
  $$\Delta_n \Sigma_n \leq 0$$

D. Explicit Results for Low Dimensions

The authors derive specific, simplified conditions for small $n$:

• Two Angles ($n=2$): They establish a region-based comparison for $A(x,y)$ and $B(x,y)$. A notable corollary is the sharp inequality:
  $$\sin(\pi x) \leq \sin(2\pi x(1-x))$$
  with equality if and only if $x = 1/2$.
• Three Angles ($n=3$): They provide a complex condition involving $\delta = y(1-2x) + z(1-2y)$ and the sign of $(2z-1)$ to determine which bound is tighter.
• Reductions: The paper presents several corollaries reducing the $n$-angle results to specific two-angle scenarios (e.g., setting $y=x$, $z=y$, etc.), yielding piecewise-defined sharp bounds.

E. Global Sharp Bound (Corollary 4.10)

By combining the results for $n=2$ and $n=3$, the authors derive a piecewise sharp upper bound for $\sin(\pi x)$ on $(0, 1)$:
$$\sin(\pi x) \leq \begin{cases} \sin(2\pi x(1-x)) & \text{if } x \in (0, 1/3] \\ \sin\left(\frac{\pi}{3}(1+2x)\right) & \text{if } x \in [1/3, 1) \end{cases}$$
This result demonstrates that no single analytic expression provides the sharpest bound across the entire domain; the optimal bound switches at $x=1/3$.

4. Significance

• Methodological Innovation: The paper successfully bridges the gap between Gamma function theory and trigonometric inequalities, offering a robust alternative to standard log-concavity arguments.
• Resolution of Ambiguity: It resolves the ambiguity of "which bound is better" by providing exact, computable algebraic criteria rather than asymptotic or numerical approximations.
• Sharpness: The results are "sharp" in the sense that the bounds are tightest possible within the framework of the chosen parameterizations, and the switching points between bounds are precisely identified.
• Future Directions: The authors suggest that this framework can be extended to $q$-analogues using the $q$-Gamma function and $q$-sine functions, opening a new avenue for research in $q$-calculus inequalities.

In summary, this paper advances the theory of trigonometric inequalities by providing a unified, Gamma-function-based derivation of weighted sine products and solving the critical problem of comparing competing upper bounds through rigorous algebraic analysis.