Hermite-Hadamard Inequalities via Gateaux Derivatives in Banach Spaces

Abstract This paper establishes a new framework for Hermite-Hadamard type inequalities by employing Gateaux derivatives and the Bochner integral. By extending the classical scalar results to real Banach spaces, we provide sharp error bounds for trapezoidal and midpoint type functional approximations. Under the assumption that the magnitude of the directional curvature $$|\varphi ^{\prime }|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>φ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is convex, we derive deterministic error estimates with an optimal constant of 1/8. Furthermore, we demonstrate the practical utility of these results through applications in nonlinear optimization search paths, quadratic growth characterization, and the stability analysis of energy potentials in nonlinear elasticity. These results remain valid in infinite-dimensional Banach spaces and therefore extend the classical Hermite-Hadamard framework beyond the usual real-variable setting. An illustrative example on the space $$\ell ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> demonstrates how the proposed approach provides effective error bounds in infinite-dimensional optimization problems.