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 σ(A) and σ(B). 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 f(ts) ≤ f(t)f(s). Under these specified conditions, we establish the following inequality: for all 0 ≤ ] ≤ 1, this outcome highlights the intricate relationship between the numerical range of the expression g](A)Xh1−] when transformed by the convex function f and the norm of X. Importantly, this inequality holds true for a broad range of values of ]. Furthermore, we provide supportive examples to validate these results.
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