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In order to prove the first assertion it suffices to show that Re ( 〈 ∂ ρ ( w ), w − z 〉 ) − ρ ( w ) > 0 for any w ∈ D ¯, z ∈ D. (9.12)Indeed, one first observes that if D is strictly convex and sufficiently smooth then Re 〈 ∂ ρ ( w ), w − z 〉 > 0 for any w ∈ D ¯ ∖ { z } (see [20] for the proof of this fact) so that Re 〈 ∂ ρ ( w ), w − z 〉 is non-negative in D ¯ and it vanishes only at w = z.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com