TY - JOUR

T1 - Sharp real-part theorems for high order derivatives

AU - Kresin, G.

AU - Maz'ya, V.

N1 - Funding Information:
The research of the first author was supported by the KAMEA program of the Ministry of Absorption, State of Israel, and by the Ariel University Center of Samaria. The second author was partially supported by the Scandinavian Network “Analysis and Applications.”

PY - 2012/2

Y1 - 2012/2

N2 - We obtain a representation for the sharp coefficient in an estimate of the modulus of the nth derivative of an analytic function in the upper half-plane ℂ +. It is assumed that the boundary value of the real part of the function on ∂ℂ + belongs to L p. This representation is specified for p = 1 and p = 2. For p = ∞ and for derivatives of odd order, an explicit formula for the sharp coefficient is found. A limit relation for the sharp coefficient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. It is assumed that the boundary value of the real part of the function belongs to L p. The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in ℂ +. As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary. Bibliography: 8 titles.

AB - We obtain a representation for the sharp coefficient in an estimate of the modulus of the nth derivative of an analytic function in the upper half-plane ℂ +. It is assumed that the boundary value of the real part of the function on ∂ℂ + belongs to L p. This representation is specified for p = 1 and p = 2. For p = ∞ and for derivatives of odd order, an explicit formula for the sharp coefficient is found. A limit relation for the sharp coefficient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. It is assumed that the boundary value of the real part of the function belongs to L p. The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in ℂ +. As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary. Bibliography: 8 titles.

UR - http://www.scopus.com/inward/record.url?scp=84857141809&partnerID=8YFLogxK

U2 - 10.1007/s10958-012-0679-5

DO - 10.1007/s10958-012-0679-5

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AN - SCOPUS:84857141809

SN - 1072-3374

VL - 181

SP - 107

EP - 125

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

IS - 2

ER -