A Review on Kobers Trigonometric and Hyperbolic Inequalities in Optimization and Control Application

Abstract

Kober’s trigonometric and hyperbolic inequalities form a significant class of functional inequalities that provide sharp bounds for nonlinear expressions involving sine, cosine, hyperbolic sine, and hyperbolic cosine functions. These inequalities, originally established to refine classical analytic results, have found widespread application in modern optimization and control theory. In control system analysis, such inequalities are instrumental in deriving stability conditions, particularly in Lyapunov-based methods and the analysis of time-delay and nonlinear systems. Moreover, they serve as essential tools in the design of robust and adaptive controllers, where bounding nonlinearities is crucial for ensuring system performance and convergence. In optimization problems involving trigonometric constraints or cost functions, Kober-type inequalities aid in the convexification and approximation of non-convex terms, thereby enabling tractable solutions. This work presents an overview of Kober’s inequalities involving trigonometric and hyperbolic functions and explores their theoretical foundations, refinements, and applications in control system design and constrained optimization.

Introduction

Hermann Kober was a mathematician noted for introducing refined inequalities involving trigonometric and hyperbolic functions, known as Kober-type inequalities. These inequalities, such as

sin?xx<cos?x2for 0<x<π\frac{\sin x}{x} < \cos \frac{x}{2} \quad \text{for } 0 < x < \pixsinx?<cos2x?for 0<x<π

and its hyperbolic counterpart, provide tighter bounds than classical inequalities like Jordan’s, offering improved precision in mathematical analysis.

Applications:

• Control Systems: Kober inequalities help estimate bounds for nonlinear functions appearing in system stability analysis via Lyapunov functions, leading to stronger stability conditions and better error control. They are particularly useful for systems with delays or fractional order dynamics.

• Optimization: By bounding nonlinear trigonometric constraints, Kober inequalities enable convex approximations of otherwise non-convex problems, facilitating solution via modern optimization methods like interior-point or Lagrangian techniques.

• Signal Processing and Filtering: These inequalities assist in error estimation where kernel functions involve sine or hyperbolic sine.

Mathematical Techniques:

• Derived using Taylor expansions, convexity, monotonicity, integral mean value theorems, and fractional integrals.

• Extend classical inequalities by providing sharper bounds on expressions involving sin?x/x\sin x / xsinx/x, sinh?x/x\sinh x / xsinhx/x, and related functions.

Key Results:

• Kober inequalities improve Lyapunov decay estimates, yielding better system performance.

• They help in the design of controllers with enhanced robustness, energy efficiency, and transient response.

• Fractional Kober integrals model memory-dependent systems, expanding control law design capabilities.

Significance:

Kober inequalities bridge abstract mathematical theory and practical engineering by refining classical bounds, enhancing stability analysis, and enabling efficient handling of nonlinearities in control and optimization. They continue to inspire new inequalities and are foundational tools in modern applied mathematics and engineering.

References

[1] H. Kober, Approximation by integral functions in the complex domain, Trans. Amer. Math. Soc. Vol. 56, No. 1, pp. 7 - 31, 1944. [2] Kober, H.“Über einige Mittelwerte mit Anwendungen auf Differential- und Integral gleichungen” Journal für die reine und angewandte Mathematik.Crelle\'s Journal, (1957). [3] D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, Berlin 1970. [4] Boris Polyak, Pavel Shcherbakov, Lyapunov Functions: An Optimization Theory Perspective , Volume 50, Issue 1, July 2017, Pages 7456-7461 [5] Barkat Ali Bhayo and J´ozsef S´andor, On Jordan’s and Kober’s Inequality, Acta Et Commentationes Universitatis tartuensis de Mathematica, Vol. 20, No. 2, Dec. 2016 [6] J. S´andor, On new refinements of Kober’s and Jordan’s trigonometric inequalities, Notes Number Theory Discrete Math. Vol. 19, No. 1, pp. 73 - 83, (2013). [7] Boyd, S., Vandenberghe, L.“Convex Optimization” (2004) [8] Sándor, J. “On Some Trigonometric and Hyperbolic Inequalities”Journal of Inequalities in Pure and Applied Mathematics, (2011). [9] Klén, R., Vuorinen, M., Zhang, X.“Inequalities for the Generalized Trigonometric and Hyperbolic Functions” Journal of Approximation Theory, (2013). [10] Dragomir, S.S.“Some Inequalities for Trigonometric and Hyperbolic Functions and Their Applications in Optimization”(2014). [11] Bagul, Y.J., and Panchal, S.K. “Certain Inequalities of Kober and Lazarevi? Type” RGMIA Research Report Collection, (2018). [12] Ames et?al., Control Barrier Functions: Theory and Applications” A comprehensive survey on inequality-driven techniques in control via barrier and Lyapunov functions\" (2019).

Copyright

Copyright © 2025 D. P. Wagh, S. P. Barhate, K. N. Chavan, S. S. Borwar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.