[Paper Review] Sharpening and generalizations of Carlson's double inequality for the arc cosine function

Bai‐Ni Guo, Feng Qi|arXiv (Cornell University)|Feb 18, 2009

Mathematical Inequalities and Applications4 citations

TL;DR

This paper sharpens and generalizes Carlson's double inequality for the arc cosine function by analyzing the monotonicity of a parameterized function $ f_{a,b}(x) = \frac{(1+x)^b}{(1-x)^a} \arccos x $. It establishes precise conditions on parameters $ a $ and $ b $ for which the function is increasing, decreasing, or has unique extrema, leading to improved and best-possible constant bounds in inequalities involving $ \arccos x $ on $ (0,1) $. The key contribution is a refined double inequality with optimal constants, extending and sharpening prior results.

ABSTRACT

In this paper, we sharpen and generalize Carlson's double inequality for the arc cosine function.

Motivation & Objective

• To refine and extend Carlson's double inequality for $ \arccos x $ on $ (0,1) $ by introducing a parameterized function $ f_{a,b}(x) = \frac{(1+x)^b}{(1-x)^a} \arccos x $.
• To determine exact conditions on parameters $ a $ and $ b $ for which $ f_{a,b}(x) $ is strictly increasing, decreasing, or has unique extrema.
• To derive new double inequalities for $ \arccos x $ with best-possible constants by analyzing the extremal behavior of $ f_{a,b}(x) $.
• To prove that the constants in the refined inequalities are optimal through limit analysis and extremal point characterization.

Proposed method

• Analyzes the monotonicity of $ f_{a,b}(x) = \frac{(1+x)^b}{(1-x)^a} \arccos x $ via differentiation and sign analysis of its derivative $ f_{a,b}'(x) $.
• Introduces auxiliary functions $ g_{a,b}(x) $, $ g_{a,b}'(x) $, and $ h(x) $ to study the convexity and monotonicity of the derivative components.
• Uses the second derivative $ g_{a,b}''(x) $ and sign analysis of $ h(x) $ to prove that $ g_{a,b}'(x) $ is increasing, enabling precise monotonicity classification.
• Derives critical points $ x_1 $ and $ x_2 $ as roots of a quadratic in $ h(x) $, which correspond to extrema of $ f_{a,b}(x) $, and uses them to bound $ f_{a,b}(x) $.
• Applies limit analysis as $ x \to 0^+ $ and $ x \to 1^- $ to determine asymptotic behavior and establish extremal bounds.
• Uses the strict decrease of $ F_{1/2,1/2,2\sqrt{2}}(x) = \frac{2\sqrt{2} + (1+x)^{1/2}}{(1-x)^{1/2}} \arccos x $ to prove the best-possible constants in the refined inequality.

Experimental results

Research questions

• RQ1 Under what conditions on $ a $ and $ b $ is the function $ f_{a,b}(x) = \frac{(1+x)^b}{(1-x)^a} \arccos x $ strictly decreasing on $ (0,1) $?
• RQ2 When does $ f_{a,b}(x) $ have a unique maximum or minimum in $ (0,1) $?
• RQ3 What are the optimal constants in the double inequality $ \frac{6(1-x)^{1/2}}{2\sqrt{2} + (1+x)^{1/2}} < \arccos x < \frac{C(1-x)^{1/2}}{2\sqrt{2} + (1+x)^{1/2}} $?
• RQ4 How can the right-hand side of Carlson's inequality be generalized and sharpened using parameterized bounds?
• RQ5 What is the necessary and sufficient condition on $ b $ for the inequality $ \arccos x < 2^{b+1/2} \frac{(1-x)^{1/2}}{(1+x)^b} $ to hold on $ (0,1) $?

Key findings

• The function $ f_{a,b}(x) $ is strictly decreasing on $ (0,1) $ if and only if $ b \leq \frac{2}{\pi} - a $ and $ a \leq \frac{1}{2} $.
• The function $ f_{a,b}(x) $ is strictly increasing if $ (a,b) \in \left\{ \frac{2}{\pi} - a \leq b \leq a - \frac{4}{\pi^2} \right\} \cup \left\{ \frac{1}{2} \leq a \leq b + \frac{1}{3} \right\} \cup \left\{ \frac{1}{3} < a - b < \frac{4}{\pi^2}, a + b \geq \frac{2(a-b)^{3/2}}{\sqrt{4(a-b)-1}} \right\} $.
• When $ \frac{1}{3} < a - b < \frac{4}{\pi^2} $, $ f_{a,b}(x) $ has a unique maximum if $ \frac{2}{\pi} - b < a \leq \frac{1}{2} $, and a unique minimum if $ \frac{1}{2} < a \leq \frac{2}{\pi} - b $.
• The double inequality $ \frac{6(1-x)^{1/2}}{2\sqrt{2} + (1+x)^{1/2}} < \arccos x < \frac{\left(\frac{1}{2} + \sqrt{2}\right)\pi (1-x)^{1/2}}{2\sqrt{2} + (1+x)^{1/2}} $ holds on $ (0,1) $, and the constants 6 and $ \left(\frac{1}{2} + \sqrt{2}\right)\pi $ are best possible.
• The right-hand side inequality $ \arccos x < 2^{b+1/2} \frac{(1-x)^{1/2}}{(1+x)^b} $ holds if and only if $ b \geq \frac{1}{6} $, and the reversed inequality holds if and only if $ b \leq \frac{2}{\pi} - \frac{1}{2} $.
• When $ 16ab(b-a) + (a+b)^2 > 0 $ and $ x_1 > 0 $, the upper bound for $ f_{a,b}(x) $ is $ \frac{(1+x_1)^{b+1/2}(1-x_1)^{1/2-a}}{a+b + (a-b)x_1} $, and when $ x_2 \in (0,1) $, the lower bound is $ \frac{(1+x_2)^{b+1/2}(1-x_2)^{1/2-a}}{a+b + (a-b)x_2} $.