Starlike and convexity properties for p-valent solutions of the Shah differential equation |
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Author |
m.m.sheremeta@gmail.com, yurkotrukhan@gmail.com
Ivan Franko National University of Lviv, Lviv, Ukraine
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Abstract |
The starlikeness and the convexity in the unit disc
and the growth of an entire function
f(z)=zp+∑∞n=p+1fnzn, p∈N,
satisfying the differential equation
z2w″
(\beta_0,\,\beta_1,\,\gamma_0,\,\gamma_1,\,\gamma_2 are complex parameters) are studied.
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Keywords |
starlikeness; convexity; entire function; differential equation
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DOI |
doi:10.15330/ms.48.1.14-23
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Reference |
1. G.M. Goluzin, Geometric theory of functions of a complex variable, M.: Nauka, 1966 (in Russian). Engl.
transl.: AMS: Translations of Mathematical monograph, 26 (1969), 676p.
2. W. Kaplan, Close-to-convex schlicht functions, Michigan Math.J., 1 (1952), №2, 169–185. 3. S.M. Shah, Univalence of a function f and its successive derivatives when f satisfies a differential equation II, J. Math. Anal. Appl., 142 (1989), №2, 422–430. 4. Z.M. Sheremeta, The properties of entire solutions of one differential equation, Diff. uravnieniya, 36, (2000), №8, 1045–1050. Engl. transl.: Diff. Equat., 36 (2000), №8, 1155–1161. 5. Z.M. Sheremeta, Close-to-convexity of entire solutions of a differential equation, Mat. method. and fiz.- mech. polya, 42 (1999), №3, 31–35. (in Ukrainian) 6. Z.M. Sheremeta, On entire solutions of a differential equation, Mat. Stud., 14 (2000), №1, 54–58. 7. Z.M. Sheremeta, On the close-to-convexity of entire solutions of a differential equation, Visn. Lviv Un-ty. Mech.Math., (2000), №58, 54–56. 8. Z.M. Sheremeta, M.M. Sheremeta, Close-to-convexity for entire solutions of a differential equation, Diff. uravnieniya, 38 (2002), №4, 477–481. Engl. transl.: Diff. Equat., 38 (2002), №4, 496–501. 9. Z.M. Sheremeta, M.M. Sheremeta, Convexity of entire solutions of a differential equation, Mat. method. and fiz.-mech. polya, 47 (2004), №2, 186–191. (in Ukrainian) 10. Ya.S. Mahola, M.M. Sheremeta, Properties of entire solutions of a linear differential equation of n-th order with polynomial coefficients of n-th degree, Mat. Stud., 30 (2008), №2, 153–162. 11. Ya.S. Mahola, M.M. Sheremeta, Close-to-convexsity an entire solution of a linear differential equation with polynomial coefficients, Visnyk of the Lviv Univ. Series Mech-Math., (2009), №70, 122–127. (in Ukrainian) 12. Ya.S. Mahola, M.M. Sheremeta, On the properties of entire solutions of linear differential equations with polynomial coefficients, Mat. metody and fiz.-mech. polya, 53 (2010), №4, 62–74 (in Ukrainian). Engl. transl.: J. Math. Sci. 181 (2012), №3, 366–382. 13. S. Owa, On certain classes of p-valent functions with negative coefficients, Simon Stevin, 59 (1985), 385–402. 14. R.M. El-Ashwah, M.K. Aouf, A.O. Moustafa, Starlike and convexity properties for p-valent hypergeometric functions, Acta Math. Univ. Comenianae, 79 (2010), №1, 55–64. |
Pages |
14-23
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Volume |
48
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Issue |
1
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Year |
2017
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Journal |
Matematychni Studii
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Full text of paper | |
Table of content of issue |