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Let Ω be an n‐dimensional convex domain, and let ν ∈ [0, 1/2]. For all f ∈ H01(Ω) we prove the inequality where δ = dist (x, ∂ Ω), δ0 = sup δ. The factor λν2 is sharp for all dimensions, λν being the first positive root of the Lamb type equation Jν(λν)+ 2λν Jν′(λν) = 0 for Bessel's functions. In particular, the case ν= 0 with λ0= 0,940 ... presents a new sharp form of the Hardy type inequality due to Brezis and Marcus, while in the case ν = 1/2 with λ1/2 = π/2 we obtain a unified proof of an isoperimetric inequality due to Poincaré for n = 1, Hersch for n = 2 and Payne and Stakgold for n ≥ 3. A generalization, when the latter integral is replaced by the integral ∫Ω|f|2 / δ2‐mdx, m > 0, is proved, too. As a special case, we obtain the sharp inequality where jν is the first positive zero of Jν.

Abstract. Let U(λ) denote the family of analytic functions f (z), f (0) = 0 = f (0) − 1, in the unit disk D, which satisfy the condition z/f (z) 2 f (z) − 1 < λ for some 0 < λ ≤ 1. The logarithmic coefficients γ n of f are defined by the formula log(f (z)/z) = 2 ∞ n=1 γ n z n . In a recent paper, the present authors proposed a conjecture that if f ∈ U(λ) for some 0 < λ ≤ 1, then |a n | ≤ n−1 k=0 λ k for n ≥ 2 and provided a new proof for the case n = 2. One of the aims of this article is to present a proof of this conjecture for n = 3, 4 and an elegant proof of the inequality for n = 2, with equality for f (z) = z/[(1 + z)(1 + λz)]. In addition, the authors prove the following sharp inequality for f ∈ U(λ):where Li 2 denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of S.

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We consider the Fekete-Szegö problem with real parameter λ for the class Co(α) of concave univalent functions.