Abstract
We introduce the notion of order generalised gradient, a generalisation of the notion of subgradient, in the context of operator-valued functions. We state some operator inequalities of Hermite-Hadamard and Jensen types. We discuss the connection between the notion of order generalised gradient and the Gâteaux derivative of operator-valued functions. We state a characterisation of operator convexity via an inequality concerning the order generalised gradient.
Highlights
1 Background Convex functions play a crucial role in many fields of mathematics, most prominently in optimisation theory
To the case of real-valued functions, the operator convexity can be characterised by some operator inequalities
In Section, we state the connection between the order generalised gradient and Gâteaux derivative of operator-valued functions
Summary
Convex functions play a crucial role in many fields of mathematics, most prominently in optimisation theory. In Section , we state the connection between the order generalised gradient and Gâteaux derivative of operator-valued functions. For any convex function defined on a segment [x, y] ⊂ X, we have the Hermite-Hadamard integral inequality 4 Inequalities involving order generalised gradients We start this section by the following definition.
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