UNIVALENCE FOR CONVOLUTIONS

The radius of univalence is found for the convolution f,# of functions f E S (normalized univalent functions) and g (5 C (close-to-convex functions). A lower bound for the radius ofunivalence is also determined when f and g range over all of S. Finally, a characterization ofC provides an inclusion relationship.


INTRODUCTION.
Denote by S the family consisting of functions f(z) z + that are analytic and univalent in A {z: zl < 1} and by K,S*, and C the subfamilies of functions that are, respectively, convex, starlike, and close-to-convex in A. It is well known that K C S*c C c S. The convolution oftwo power series and g(z) E bnz" is defined as the power series n-----0 n=0 y(z) E ,.b.z". n----O The Kvebe function k(z)= z/(1z) often plays an extremal role in the family S.This enables us to show it to be extreme in many convolution problems.For example, the modulus ofthe nth coefficient for f,g, f and g in S, is n 2 and is attained when f g k.Similarly, f*g takes its maximum and minimum on the circle z r when f g k.
A question was raised in [4] as to whether rnin Re (f,)(z) rnin Re (f,k)(.z)rnin Ref when f and g are taken over all of S. The classical rotation theorem for f (5 S leads to the sharp result that Ref'(z) > 0 when zl < sin(r/8).This was generalized in [4] to Re (f,9)(z) > 0 for z < s/n (7r/8) when f (5 S and g (5 S*, but could not be extended to g (5 S or even to g (5 C. In particular, functions f,g(5 C were found for which Re (l,9)(z) < 0 at some point zo,' 01 < ,(/8).In this note, we investigate the radius of umvalence for f,g, f and g in S For f E S and g k, the Koebe function, f,g s umvalent m the dsk z <2-We prove that g-k can be replaced by any g E C, but we cannot settle f ths extends to arbitrary g S. We do show, however, that f.g is univalent for at least z < .8(2x/) 2. MAIN RESULTS.
THEOREM 1. IffSandgC, then f.gs univalent m Izl <2-x,/.The results sharp PROOF.It s well known that f is convex in z < r if and only if zf' is starlike in z < r and that the radius of convexity of S s 2-x/.Thus, f.k zf' has radius of starlikeness (and hence radius of umvalence) at least 2 x/, the radms of convexity for f S Since (,:), (:,), +4+ o at (2 x/S) (l_z)a the radius ofunivalence off,g for f, g E S can be no greater than r 2 x/-.
When f S, we have f(az)/a K for a 2 x/.Hence, by a theorem of Ruscheweyh and Sheil-Small [3], if f S and g C then s() , g(z) e C c S.
Thus, f,g is univalent for z < 2 V/-, and the proof is complete.
In our next theorem, we replace C with S in the hypothesis and this leads to a weaker conclusion.
PROOF.The upper bound was found in Theorem 1. Krzyz determined the radius of close-to-convexity for S to be to 0.80 +.Since f(az)/a e K, a 2-f, and g(toz)/to _ C, (toz we have from the Ruscheweyh and Sheil-Small theorem [3] that Y( z----2,-77-0 C, which shows that f,g is univalent for z < t0(2-x/-).This furnishes us with the lower bound, and the proof is complete.
Though we are unable to prove that r0 2- in Theorem 2, the lower bound on r0 most certainly can be improved.Ruscheweyh defined the family M consisting of normalized functions f by M--{f:f,gO;g_S*,O< Izl <1}.
He proved the proper inclusions C c M C S and that f,g M for f K and g M [2].Hence, if tx is the largest value for which g(tlz)/tl M when g S, methods identical to those ofTheorem 2 show that f,g is univalent in z < tl (2 /') for f, g E S. Unfortunately the value of tl, the radius of"M-ness" for S, is unknown.

A CHARACTERIZATION OF C.
The inclusion C c M is not obvious and was proved by Ruscheweyh using his duality principle [2].Our final result is a characterization of C that leads to a more elementary proof that C c M.
We make use of a result found in UNIVALENCE FOR CONVOLUTIONS 203 THEOREM 3. A function f E C if and only f to each E S* we may associate an h S* for which Re f*g > O, z a__ A. h PROOF.To show that the condition is sufficient for f to be in C, we choose zf' y(z) z/(1 z) e S*.Then Re -Re ---> 0, which means that f E C.
On the other hand, f fC we can find a =S* for which Rezf'/ >0 Set F(z) zf'(z)/(z).Then for y E S* there corresponds E K such that z'= y.Note that f*9 zf', ,F and that h , S*.By Lemma A, ,Fq Re Re >0, and the proof is complete COROLLARY.C c M PROOF.Since Re f*h > 0 = f,g :/: O, the result follows from Theorem 3.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: [3].LEMMA 3. If K, S*, and F is analytic with ReF > 0 for z A, then Re-2--> O.