Sharp inequalities for sine and cosine on complex circles and applications

Abstract

We investigate sharp inequalities for the functions | sin z| and | cos z| along circles in the complex plane. Building on recent results of Qi, we establish precise bounds for the quantities | sin(reiθ)| − | cos(reiθ)| and | sin(reiθ) − cos(reiθ)|.
We show that their behavior undergoes a phase transition governed by a unique critical parameter r0 defined by cos(2r0) =2r0. As an application, we completely resolve several open problems posed by Bagul and Chesneau concerning doublesided inequalities for trigonometric and hyperbolic functions.
In particular, we prove that
sin(kx)/kx+ ksin x/x> 1 + k cos x, holds if and only if k ∈ (0, 2), and that sinh(qx)/qx+ qsinh x/x> 1 + q cosh x
holds if and only if q ≥ 2. We further obtain weighted extensions of these inequalities. Our approach combines complexanalytic techniques with sharp real-variable inequalities and reveals new connections between classical inequalities and the geometry of analytic functions on circles.