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Let μ be a finite positive Borel measure on the open unit disc D and H a set of all analytic functions on D . For each a in D , put r(μ, a) = sup |f (a)|2 where f ∈ H and ∫ D |f |2dμ 1 . Unless the support set of μ is a finite set, ∫ D r(μ, a)dμ(a) = ∞ . However sup z∈D ∫ Dt(z) r(μ, a)dμ(a) < ∞ may happen where Dt(z) denotes the Bergman disc in D . We study when this is possible. When ν is a discrete measure such that dν = ∑ a∈A s(μ, a)δa , sup z∈D ∫ Dt(z) r(μ, a)dν(a) = sup z∈D ∑ a∈A∩Dt(z) 1. Under some condition on μ , we show that sup z∈D ∫ Dt(z) r(μ, a)dν(a) < ∞ for a finite positive Borel measure ν on D if and only if (ν,μ) -Carleson inequality is valid. Mathematics subject classification (1991): 46E15, 30A98, 47B99.

Riesz's functions and Carleson inequalities