Inequalities for analytic functions associated with hyperbolic cosine function

In this paper, we investigate the geometric properties of a specific subclass of analytic functions satisfying  the condition f' (z) ≺ cosh(√ z) meaning that the function f '(z) is subordinate to the function cosh(√ z). Also, we focus on deriving sharp inequalities for Taylor coefficients, particularly for b2 and the modulus of the second derivative f''(z). Utilizing the Schwarz lemma, both on the unit disc and on its boundary, we provide essential insights into the distortion and growth behaviors of these functions. The paper demonstrates the sharpness of these inequalities through extremal functions and applies the Julia–Wolff lemma to establish boundary behavior results. These findings contribute significantly to the understanding of the analytic functions associated with the hyperbolic cosine function, with potential applications in geometric function theory. It is considered that the extremal functions obtained in this study could be potential hyperbolic activation functions in neural network architectures. This perspective builds a conceptual bridge between geometric function theory and artificial intelligence, indicating that insights from complex analysis can inspire the development of more effective and theoretically grounded activation mechanisms in deep learning. Empirical evaluation of architectures built with novel activation functions may be considered as potential future work.