UNIVALENT FUNCTIONS DEFINED BY RUSCHEWEYH DERIVATIVES

We study some radii problems concerning the integral operator z F(z)y+l uY-I f(u) du zy o for certain classes, namely K and M (a), of univalent functions defined by Ruscheweyh n n derivatives. Infact, we obtain the converse of Ruscheweyh’s result and improve a result of Goel and Sohi for complex by a different technique. The results are sharp.


INTRODUCTION k
Let S denote the class of functions of the form f(z) z + E a,. z which are regular in the unit disc U {z zl < i}. and the operation (*) stands for the convolution or Hadamard product of the power series.
Ruscheweyh [1] introduced the classes K and showed, via the inclusion relation n Kn+1 K that the functions in these classes are starlike of order 1/2 and hence are n univalent.He also observed that Following A1-Amiri [2], we also refer to Dnf as the nth order Ruscheweyh derivative of f.
A function f of S is said to belong to the class M (), 0 < Goel and Sohi [3] introduced the classes M (c) and showed, via the inclusion n relation Mn+ l(a) Mn(a), that the functions in these classes are univalent.
Ruscheweyh [1] proved that the function F defined by z F(z) K and y is a complex number such that Re(y)>(n-l)/2.Goel and n n Sohi [3] obtained an analogous result for the class M ().Conversely, they [3, n Theorem 4] determined the radius of the disc in which f M () when F g M () and n n y is a real number such that y > -i.
In the present paper we obtain the converse of Ruscheweyh's [i] result.We also obtain the above mentioned result of Goel and Sohi [3, Theorem 4], by using a different technique, for complex y The results are shown to be sharp.

PRELIMINARY LEMMA.
Let P denote the class of functions of the form P(z) i + l bkzk which are o k=l regular in U and satisfy Re {p(z)} > 0 for z & U.

MAIN RESULTS.
In the following theorem we study the converse of Ruscheweyh's [I] result THEOREM 3.1 Let y be a complex number such that Re(y) > -i.If F e K n then the function f defined by z ) is given by Lemma 2.1.The result is sharp.
For the existence of the integral in (3.1), the power represents principle branch.
We note that the integral operator under consideration can also be written as i t Y-I f(tz) dt F(z) (y+l) o which solves the question of principal branch.
PROOF.It is easy to verify the identity z(DnF(z)) Also, from the definition of F it can be verified that z(D n F(z))' Since F g K there exists a function p in P such that  The required result now follows by using Lemma 2.1.
In the following theorem, we obtain the converse of the result of Goel and Sohi [3, Theorem 2] for complex y.THEOREM 3.2.Let F e M (s) and y be a complex number such that Re(y)> -i.ez + (l-a) z i_--).
Goel and Sohi [3, Theorem proved that, if f M (e), then the function F n defined by (3.1) also belongs to Mn(e) provided that Re(y) > -i.In this direction, the following theorem provides a better result for suitable choices of y THEOREM 3.3.If f M (s) and y is a real number such that -i < y < n+l, then the n function F defined by (3.1) belongs to Mn+l(e). PROOF.Since and, by the definition of F, M (e), the above inequality leads us to n Dn+2F(z) Hence, F e Mn+ l(a).
ACKNOWLEDGEMENT.The authors are thankful to the referee for his valuable suggestions about the earlier version of tM[s paper.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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k=2A
function f of S is said to belong to the class K if n {Dn+l.i f(.z,).}>where z U, n e N O {0,i,2,...}, < R*.The sharpness of the result follows easily by taking the function F defined by (l+z.D n+l F(z)