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Let X be a complex Banach space and p a real number with p 1 . We give a necessary and sufficient condition for complex numbers a, b and real numbers λ , μ and ν in order that the inequality ‖ax + by‖p λ ‖x‖ p μ + ‖y‖p ν holds for every x, y ∈ X . Mathematics subject classification (2010): 26D15, 26D20.

Refined generalizations of the triangle inequality on Banach spaces