On some inequalities for weighted products and ratios of the sine function

Technical Summary: Inequalities for Weighted Products and Ratios of the Sine Function

Problem Statement
The paper addresses the problem of establishing sharp inequalities for weighted products and ratios of sine functions. Specifically, it investigates the classical sine-concavity inequality:
$$\prod_{i=1}^n \sin^{\alpha_i}(\theta_i) \leq \sin^\alpha \left( \sum_{i=1}^n \frac{\alpha_i}{\alpha} \theta_i \right)$$
where $\alpha = \sum \alpha_i$ and $\theta_i \in (0, \pi)$. While this inequality is well-known via Jensen's inequality applied to the concave function $\log \sin$, the authors observe that the standard proof utilizes only one representation of the sine function. The paper seeks to exploit the reflection identity $\sin(\pi x) = \sin(\pi(1-x))$ to generate a broader family of bounds. Additionally, the paper addresses the derivation of sharp lower bounds for ratios of the form $\prod \sin^{\alpha_i}(\pi x_i)$, a direction not previously treated with this level of generality.

Methodology
The core methodology is a "reflection-substitution technique" combined with Hölder's inequality and its reverse. The approach relies on the integral representation of the Beta function, $B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt$, and the reflection formula $B(1-x, x) = \pi / \sin(\pi x)$.

Key methodological steps include:

1. Independent Reflection Choices: For each coordinate $x_i \in (0, 1)$, the term $\sin(\pi x_i)$ can be represented via the Beta integral using either the exponent pair $(x_i, 1-x_i)$ or $(1-x_i, x_i)$. The authors exploit the freedom to choose these representations independently for all $n$ coordinates.
2. Application of Hölder's Inequality:
   • Upper Bounds: By applying the standard Hölder inequality to the $2^n$ possible integral representations of the weighted product, the authors derive a family of upper bounds.
   • Lower Bounds: By applying the reverse Hölder inequality (requiring specific sign conditions on the weights) to ratios of sines, they derive a corresponding family of lower bounds.
3. Classification of Bounds: The authors define "reflection sets" (subsets of indices where the complement $1-x_i$ is used) to parameterize these bounds. They demonstrate that while there are $2^n$ representations, they collapse into $2^{n-1}$ distinct bounds due to the symmetry $\sin(\pi M) = \sin(\pi(1-M))$.
4. Comparison Criteria: The comparison between the various bounds within a family is reduced to determining the sign of linear combinations of the variables $(2x_i - 1)$. This allows for the explicit identification of the sharpest bound for any given configuration of variables.

Key Contributions and Results

• Generalization of Upper Bounds: The paper establishes that for $n$ weights, there exist $2^{n-1}$ distinct upper bounds for the weighted product of sines. The classical Jensen bound corresponds to one specific choice of reflection set (the empty set), while the bound from the authors' previous work [5] is identified as another member of this family.
• Novel Lower Bounds: The paper introduces $2^{n-1}$ distinct lower bounds for weighted sine ratios, a result not previously available in the literature.
• Explicit Dominance Criteria: The authors provide closed-form criteria to determine which of the $2^{n-1}$ bounds is the sharpest (minimum for products, maximum for ratios) based on the signs of linear combinations of the input variables.
• Complete Classification in Low Dimensions:
  • Two Dimensions ($n=2$): The paper fully classifies the two-dimensional case, showing that two distinct bounds exist. The dominance region is determined by the sign of $(1-2y)(2x-1)$.
  • Three Dimensions ($n=3$): The paper provides a complete analysis for $n=3$, identifying four distinct bounds. It details the dominance regions for both upper bounds and lower bounds (considering two specific weight signatures: $(+, -, -)$ and $(+, +, -)$).
• Numerical Verification: Comprehensive numerical examples are provided in tables for 2D and 3D cases, verifying that the theoretical dominance criteria correctly identify the best bound in various parameter configurations.

Significance and Claims
The paper claims that the reflection-substitution technique reveals a structural richness in sine inequalities that is overlooked by classical proofs. By recognizing that each coordinate offers two equally valid integral representations, the authors move beyond a single fixed inequality to a "selectable family of bounds."

The significance of this work lies in:

1. Unification: It unifies previous results (including the authors' earlier work) into a single framework.
2. Explicit Optimality: Unlike existence proofs, this framework allows for the explicit selection of the sharpest bound for any specific numerical configuration without trial and error.
3. Extension to Ratios: It extends the methodology to lower bounds for ratios, opening a new direction in the analysis of such expressions.
4. Scalability: While the complexity of comparison grows with dimension, the paper asserts that the general framework provides a template for $n \geq 4$, even if the explicit criteria become more elaborate.

The authors conclude that these results are directly useful in contexts where weighted sine products and ratios appear, such as in estimates for Beta and Gamma functions, as they provide a mechanism to choose the tightest possible estimate based on the specific values of the variables. They also suggest that the method may be extendable to $q$-analogues via the $q$-Gamma function.