Explicit Real-Part Estimates for High Order Derivatives of Analytic Functions

The representation for the sharp constant K _{n , p} in an estimate of the modulus of the n-th derivative of an analytic function in the upper half-plane C₊ is considered in this paper. It is assumed that the boundary value of the real part of the function on ∂C₊ belongs to L^p. The representation for K _{n , p} implies an optimization problem for a parameter in some integral. This problem is solved for p= 2 (m+ 1) / (2 m+ 1 - n) , n≤ 2 m+ 1 , and for some first derivatives of even order in the case p= ∞. The formula for K _{n , 2 ( m + 1 ) / ( 2 m + 1 - n )} contains, for instance, the known expressions for K _{2 m + 1 , ∞} and K _{m , 2} as particular cases. Also, a two-sided estimate for K _{2 m , ∞} is derived, which leads to the asymptotic formula K _{2 m , ∞}= 2 ((2 m- 1)) ²/ π+ O(((2 m- 1)) ²/ (2 m- 1)) as m→ ∞. The lower and upper bounds of K _{2 m , ∞} are compared with its value for the cases m= 1 , 2 , 3 , 4. As applications, some real-part theorems with explicit constants for high order derivatives of analytic functions in subdomains of the complex plane are described.