## Abstract

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.

Original language | English |
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Pages (from-to) | 637-652 |

Number of pages | 16 |

Journal | Computational Methods and Function Theory |

Volume | 16 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2016 |

## Keywords

- Analytic functions
- Asymptotic formula
- Explicit real-part estimates
- High order derivatives