Abstract:
It is shown that new inequalities for certain classes of entire functions can be obtained by applying the Schwarz lemma and its generalizations to specially constructed Blaschke products. In particular, for entire functions of exponential type whose zeros lie in the closed lower half-plane, distortion theorems, including the two-point distortion theorem on the real axis, are proved. Similar results are established for polynomials with zeros in the closed unit disk. The
classical theorems by Turan and Ankeny–Rivlin are refined. In addition, a theorem on the mutual disposition of the zeros and critical points of a polynomial is proved. Bibliography: 16 titles.
Citation:
V. N. Dubinin, “Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros”, Analytical theory of numbers and theory of functions. Part 21, Zap. Nauchn. Sem. POMI, 337, POMI, St. Petersburg, 2006, 101–112; J. Math. Sci. (N. Y.), 143:3 (2007), 3069–3076
\Bibitem{Dub06}
\by V.~N.~Dubinin
\paper Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros
\inbook Analytical theory of numbers and theory of functions. Part~21
\serial Zap. Nauchn. Sem. POMI
\yr 2006
\vol 337
\pages 101--112
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl184}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2271959}
\zmath{https://zbmath.org/?q=an:1117.30016}
\elib{https://elibrary.ru/item.asp?id=9305276}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2007
\vol 143
\issue 3
\pages 3069--3076
\crossref{https://doi.org/10.1007/s10958-007-0192-4}
\elib{https://elibrary.ru/item.asp?id=13546865}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34248139007}
Linking options:
https://www.mathnet.ru/eng/znsl184
https://www.mathnet.ru/eng/znsl/v337/p101
This publication is cited in the following 28 articles:
E. G. Kompaneets, L. G. Zybina, “Smirnov and Bernstein-type inequalities, taking into account higher-order coefficients and free terms of polynomials”, Probl. anal. Issues Anal., 13(31):1 (2024), 3–23
Mohd Yousf Mir, Lubna Wali Shah, Wali Mohammad Shah, “Rate of growth of polynomials non vanishing inside a circle”, B Soc Paran Mat, 42 (2024), 1
U. M. Ahanger, W. M. Shah, “Inequalities for the Derivatives of Rational Functions with Prescribed Poles and Restricted Zeros”, Vestnik St.Petersb. Univ.Math., 56:3 (2023), 392
Sergiy A. Plaksa, Vitalii S. Shpakivskyi, Frontiers in Mathematics, Monogenic Functions in Spaces with Commutative Multiplication and Applications, 2023, 161
Arnisa Rasri, Jiraphorn Somsuwan Phanwan, Narendra K. Govil, “Some Inequalities Involving the Derivative of Rational Functions”, International Journal of Mathematics and Mathematical Sciences, 2023 (2023), 1
Abdullah Mir, Adil Hussain, “Extremal problems of Turán-type for a univariate complex coefficient polynomial”, Anal.Math.Phys., 13:3 (2023)
N. A. Rather, A. Iqbal, I. A. Dar, “Inequalities for Rational Functions with Prescribed Poles”, Math. Notes, 114:4 (2023), 593–607
Nisar Ahmad Rather, Mohmmad Shafi Wani, Ishfaq Dar, “Inequalities pertaining to rational functions with prescribed poles”, Ural Math. J., 8:2 (2022), 143–152
Gradimir V. Milovanović, Abdullah Mir, Adil Hussain, “Inequalities of Turán-type for algebraic polynomials”, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 116:4 (2022)
Prasanna Kumar, Ritu Dhankhar, “On the Location of Zeros of Polynomials”, Complex Anal. Oper. Theory, 16:1 (2022)
M. Y. Mir, S. L. Wali, W. M. Shah, “Inequalities for Meromorphic Functions Not Vanishing Outside the Disk”, J Math Sci, 266:4 (2022), 526
V. N. Dubinin, “Some remarks on rotation theorems for complex polynomials”, Sib. elektron. matem. izv., 18:1 (2021), 369–376
S. L. Wali, W. M. Shah, “Bernstien type inequalities for polynomials with restricted zeros”, J Anal, 29:4 (2021), 1083
N. A. Rather, Ishfaq Dar, A. Iqbal, “Some inequalities for polynomials with restricted zeros”, Ann Univ Ferrara, 67:1 (2021), 183
N. A. Rather, Ishfaq Dar, A. Iqbal, “On a refinement of Turán's inequality”, Complex Anal Synerg, 6:3 (2020)
Adil Hussain, Abrar Ahmad, “Generalizations of some Bernstein-type inequalities for the polar derivative of a polynomial”, Ann Univ Ferrara, 66:2 (2020), 371
Abdullah Mir, “Some Inequalities for Rational Functions with Fixed Poles”, J. Contemp. Mathemat. Anal., 55:2 (2020), 105
Abdullah Mir, Imtiaz Hussain, Ajaz Wani, “A note on Ankeny–Rivlin theorem”, J Anal, 27:4 (2019), 1103
Abdullah Mir, Ajaz Wani, M. H. Gulzar, “Some inequalities concerning the polar derivative of a polynomial”, Journal of Interdisciplinary Mathematics, 21:6 (2018), 1387
Citation in format AMSBIB
Contact us: