On derivability criteria of h-Convex Functions

This study pursues two main objectives. First, we aim to generalize the Criterion of Derivability for convex functions, which posits that for a specific type of mathematical function defined on an interval, the function is convex if and only if its rate of change (first derivative) is monotonically increasing across that interval. We aim to expand this concept to encompass the realm of 'h-convexity' which generalizes convexity for nonnegative functions by allowing a function h to act on the right hand side of the convexity inequality. Additionally, we delve into the second criterion of convexity, which asserts that for a similar type of function on an interval, the function is convex if and only if its second derivative remains non-negative across the entire interval, adhering to the conventional definition of convexity. Our goal is to reinterpret this criterion within the framework of 'h-convexity'. Furthermore, we prove that if a certain non-zero function defined on the interval [0,1] is non-negative, concave, and bounded above by the identity function, then this function is fixing the end point of the interval if and only if it is the identity function.Finally, we will also provide a response in to the conjecture given by Mohammad W. Alomari (See [6]) that it is incorrect with two counterexamples.