A real part theorem for the higher derivatives of analytic functions in the unit disk. (English) Zbl 1277.31003
Summary: Let \(n\) be a positive integer. Let \(U\) be the unit disk and let \(h^p(U)\), \(p\geq 1\), be the Hardy space of harmonic functions. G. Kresin and V. G. Maz’ya [Sharp real-part theorems. A unified approach. Berlin: Springer (2007; Zbl 1117.30001)] found a representation for the function \(H_{n,p}(z)\) in the inequality
\[ \big|f^{(n)} (z)\big|\leq H_{n,p}(z)\big\| \mathfrak R (f-\mathcal{P }_{l})\big\| _{h^p(U)},\quad\mathfrak{R}f\in h^p(U),\quad z\in U, \]
where \(\mathcal{P}_l\) is a polynomial of degree \(l\leq n-1\). We determine the sharp constant \(C_{p,n}\) in the inequality \(H_{n,p}(z)\leq\frac{C_{p,n}}{(1-|z|^2)^{1/p+n}}\). This extends a recent result of D. Kalaj and M. Marković [Complex Anal. Oper. Theory 7, No. 4, 1167–1183 (2013; Zbl 1279.31001)], where only the case \(n=1\) was considered. As a corollary, an inequality for the modulus of \(n\)-th derivative of an analytic function defined in a complex domain with bounded real part is obtained. This result improves a recent result of G. Kresin and V. Maz’ya [J. Math. Sci., New York 181, No. 2, 107–125 (2012); translation from Probl. Mat. Anal. 62, 3–17 (2011; Zbl 1277.30016)].
\[ \big|f^{(n)} (z)\big|\leq H_{n,p}(z)\big\| \mathfrak R (f-\mathcal{P }_{l})\big\| _{h^p(U)},\quad\mathfrak{R}f\in h^p(U),\quad z\in U, \]
where \(\mathcal{P}_l\) is a polynomial of degree \(l\leq n-1\). We determine the sharp constant \(C_{p,n}\) in the inequality \(H_{n,p}(z)\leq\frac{C_{p,n}}{(1-|z|^2)^{1/p+n}}\). This extends a recent result of D. Kalaj and M. Marković [Complex Anal. Oper. Theory 7, No. 4, 1167–1183 (2013; Zbl 1279.31001)], where only the case \(n=1\) was considered. As a corollary, an inequality for the modulus of \(n\)-th derivative of an analytic function defined in a complex domain with bounded real part is obtained. This result improves a recent result of G. Kresin and V. Maz’ya [J. Math. Sci., New York 181, No. 2, 107–125 (2012); translation from Probl. Mat. Anal. 62, 3–17 (2011; Zbl 1277.30016)].
References:
| [1] | Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Springer, New York (1992) · Zbl 0765.31001 |
| [2] | Beck, M., Halloran, M.: Finite trigonometric character sums via discrete Fourier analysis. Int. J. Number Theory 6(1), 51–67 (2010) · Zbl 1268.11106 · doi:10.1142/S1793042110002806 |
| [3] | Colonna, F.: The Bloch constant of bounded harmonic mappings. Indiana Univ. Math. J. 38(4), 829–840 (1989) · Zbl 0677.30020 · doi:10.1512/iumj.1989.38.38039 |
| [4] | Garnett, J.: Bounded analytic functions. In: Pure and Applied Mathematics, vol. 96. Academic Press, Inc., New York (1981) · Zbl 0469.30024 |
| [5] | Khavinson, D.: An extremal problem for harmonic functions in the ball. Canad. Math. Bull. 35, 218–220 (1992) · Zbl 0776.31004 · doi:10.4153/CMB-1992-031-8 |
| [6] | Kalaj, D., Vuorinen, M.: On harmonic functions and the Schwarz lemma. Proc. Am. Math. Soc. 140, 161–165 (2012) · Zbl 1247.31002 · doi:10.1090/S0002-9939-2011-10914-6 |
| [7] | Kalaj, D., Marković, M.: Optimal estimates for the gradient of harmonic functions in the unit disk. In: Complex analysis and operator theory. doi: 10.1007/s11785-011-0187 . arXiv:1012.3153. · Zbl 1279.31001 |
| [8] | Kresin, G., Maz’ya, V.: Sharp real part theorems for higher order derivatives. J. Math. Sci. 181(2), 107–125 (2012) · Zbl 1277.30016 · doi:10.1007/s10958-012-0679-5 |
| [9] | Kresin, G., Maz’ya, V.: Sharp real-part theorems: a unified approach. In: Lecture Notes in Mathematics, vol 1903. Springer, Berlin (2007) |
| [10] | Macintyre, A.J., Rogosinski, W.W.: Extremum problems in the theory of analytic functions. Acta Math. 82, 275–325 (1950) · Zbl 0036.04503 · doi:10.1007/BF02398280 |
| [11] | Pavlović, M.: Introduction to function spaces on the disk, vol 20. Matematički Institut SANU, Belgrade (2004) · Zbl 1107.30001 |
| [12] | Ruscheweyh, S.: Two remarks on bounded analytic functions, Serdica 11(2), 200–202 (1985) · Zbl 0581.30009 |
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