Certain Inequalities Related to the Generalized Numeric Range and Numeric Radius That Are Associated with Convex Functions - Feras Bani-Ahmad; M. H. M. Rashid (Journal of Mathematics)
In this paper, we delve into the intricate connections between the numerical ranges of specific operators and their transformations using a convex function. Furthermore, we derive inequalities related to the numerical radius. These relationships and inequalities are built upon well-established principles of convexity, which are applicable to non-negative real numbers and operator inequalities. To be more precise, our investigation yields the following outcome: consider the operators A and B, both of which are positive and have spectra within the interval m,M, denoted as σA and σB. In addition, let us introduce two monotone continuous functions, namely, g and h, defined on the interval m,M. Let f be a positive, increasing, convex function possessing a supermultiplicative property, which means that for all real numbers t and s, we have fts≤ftfs. Under these specified conditions, we establish the following inequality: for all 0≤ν≤1, this outcome highlights the intricate relationship between the numerical range of the expression gνAXh1−ν when transformed by the convex function f and the norm of X. Importantly, this inequality holds true for a broad range of values of ν. Furthermore, we provide supportive examples to validate these results.