Integral means, univalent functions and circular symmetrization

Baernstein, Albert

doi:10.1007/bf02392144

Cited by 242 publications

(140 citation statements)

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“…Like other known characterizations of equality cases in geometric and integral inequalities involving symmetries or symmetrizations (see e.g. [2], [4], [6], [7], [8], [9], [10], [11], [16], [17]), the issues discussed in these theorems hide quite subtle matters. Their treatment calls for a careful analysis exploiting delicate tools from geometric measure theory.…”

Section: Up To Translations Along the Y-axis)mentioning

confidence: 96%

The perimeter inequality under Steiner symmetrization: Cases of equality

2005

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Section: Up To Translations Along the Y-axis)mentioning

confidence: 96%

The perimeter inequality under Steiner symmetrization: Cases of equality

2005

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“…Let Ω s := S 1 • S 2 (f (sD)). Since the Green function is increased by Steiner symmetrization (see [3]) and by domain expansion, we have Suppose that Φ D (r) = Φ D (s) for some 0 < r < s < 1. Then Φ D (r) = Φ D (ρ) = Φ D (s) for all ρ with r < ρ < s. We must have equality in the Steiner symmetrization inequality for the Green function.…”

Section: 3mentioning

confidence: 99%

Geometric versions of Schwarz’s lemma for quasiregular mappings

Betsakos

2011

Proc. Amer. Math. Soc.

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Abstract. We prove monotonicity and distortion theorems for quasiregular mappings defined on the unit ball B n of R n . Let K I (f ) be the inner dilatation of f and let α = K I (f ) 1/(1−n) . Let m n denote n-dimensional Lebesgue measure and c n be the reduced conformal modulus in R n . We prove that the functions r −nα m n (f (rB n )) and r −α c n (f (rB n )) are increasing for 0 < r < 1. These results can be viewed as variants of the classical Schwarz lemma and as generalizations of recent results by Burckel et al. for holomorphic functions in the unit disk.

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“…(b) For every (-oo, oo), The definitions of u*(x) and the symmetric decreasing arrangement function can be found in [2]. LEMMA 3.3 [3].…”

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confidence: 99%