Schlict solution of Briot-Bouquet diferential subordinations involving linear sums

• Ajai P. Terwase Department of Mathematics, Faculty of Physical Sciences, Plateau State University Bokkos, Nigeria
• Opoola T. O. Department of Mathematics, University of Ilorin,Ilorin, Nigeria.
• Babalola K.O. K. O. Department of Mathematics, University of Ilorin, Ilorin, Nigeria.

Keywords: Analytic function, subordination, Hadamard product, linear combination

Abstract

It is well know that many important classes of univalent functions, for example the convex and starlike functions, are related through their derivatives by functions of positive real parts. These functions plays an important part in problem solving from signal theory, moment problems and in constructing quadrature formulas among other applications. This paper focus on an important classes of an analytic function with positive real part defined by linear sums, of particular interest is its order of schlictness in the unit disc E .

 

References

Ahuja, O. P. and Silverman, H.; Classes of functions whose derivatives have positive real parts, J. Math. Anal. Appl, 138(2)(1989), 385--392.

Al-Amiri, H. S. and Reade, M. O.; On a linear combination of some expressions in the theory of univalent functions, Monatshefto fur mathematik, bf 80 (1975), 257--264.

Babalola, K. O.; On linear combination of some geometric expressions, Indian journal of Mathematics, Special Memorial, 51 (2009), 1--8.

Eienigburg, P., Miller, S. S., Mocanu, P. T. and Read, M. O.; On a Briot-Bouquet differential subordination, Rev. Roumaine Math. Pures. Appl., 29 (1984), 567--573.

Hallenbeck, D. J. and Ruscheweyh, S. B.; Subordination by convex functions, Proc. Amer. Math. Soc., 52 (1975), 191--195.

Miller, S. S. and Mocanu, P. T.; Differential subordinations thoery and applications, Monographs and Textbooks in pure and applied mathematics, Vol. 225, Marcel Dekker Inc., New York/Basel, 2000.

Miller S. S. and Mocanu, P. T.; Schlict solutions of Briot-Bouquet differential equations, Lecture Notes in Mathematics, Springer Berlin/Heidberg, 1013 (1983), 292--310.

Miller, S. S. and Mocanu, P. T.; Differential subordinations and inequalities in the complex plane, J. of differential equ., 67 (1987), 199--211.

Miller, S. S., Mocanu, P. T. and Sebb, L.; Order of starlikeness for certain integral operators, Lecture notes in Math., 1013 Springer-Verlag, Berlin, 1983.

Mocanu, P. T.; Une propriété de convexité généralisée dans la théorie de la représentation conforme, Mathematica (Cluj), 11(34) (1969), 127--133.

Padmanabhan, K. S. and Parvatham, R.; Some applications of differential surbodinations, Bull. Austria. Math. Soc., 32(2) (1985), 321--330.

Padmanabhan, K. S. and Manjini, R.; Certain applications of differential surbordinations, Pubi de L'Institutathe'Matique, 39 (1986), 107--118.

Patel, J. and Rout, S.; An application of differential subordinations, Rendiconti di Matematica Serie VII, 14 (1994), 367--384.

Ruscheweyh, S.; New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1) (1975), 109--115.

Shanmugam, T.N.; Convulotion and differential subordination, Intl. J. of Math. and Math. Sci., 12(2) (1989), 333--340.

Singh, S., Gupta, S. and Singh, S.; On a problem of univalence of functions satisfying a differential inequality, Mathematical Inequalities and Applications, 10(1) (2007), 95--98.

Singh, V., Singh, S. and Gupta, S.; A problem in the theory of univalent functions, Integral Transforms and Special Functions, 16(2) (2005), 179--186.

Terwase, Ajai P. and Maslina, D.; Certain properties of an operator involving surbordination, Probl. Anal. Issues Anal., 22 (2015), 65--72.

Whittaker, E. T. and Watson, G. N.; A Course on modern analysis, fourth ed. (reprinted), Cambridge Univ. Press, Cambridge, 1927.

Wilken, D. R. and Feng, J.; A remark on convex and starlike functions, J. London Math. Soc., 21 (1980), 287--290.

Yamaguchi, K.; On functions satisfying $Releft{frac{f(z)}{z}right}>0$, Proc. Amer. Math. Soc., 17 (1966), 588--591.