ABSTRACT
"Examining a topic that has been the subject of more than 300 articles since it was first conceived nearly 20 years ago, this monograph describes for the first time in one volume the basic theory and multitude of applications in the study of differential subordinations."
TABLE OF CONTENTS
chapter |1 pages
Rea > 0, satisfies q(U)
J 1}, by using (2.2-7) and (2.2-8) the condition of admissibility z and n
chapter |1 pages
fi[O, 1], then I zp"(z) + 4zp'(z) + 2p(z)-I < (M- Ip(z) I <
z in a )] ), ]/2. The second condition in (2.4-7) implies that If p 1 , n]
chapter 3|2 pages
APPLICATIONS OF FIRST-ORDER DIFFERENTIAL SUBORDINATIONS
3.1. First-Order Linear Differential Subordinations
chapter |1 pages
if y 1, then this theorem reduces to Robinson's differential sub-
z ]II ]a and q(z) 1-z 1-z
chapter |4 pages
If we apply this last theorem to the special relationship between the integral = f]
of Theorem 3.2j, with f] .A = zK'
chapter |2 pages
< a1/3. Therefore, by Theorem 3.4i we deduce the
I 1/4. In this case we have = if I /4, = e(l-c)z] > zp'(z)-
chapter |2 pages
a, y, o and a be real numbers satisfying
s*,g l[f,g] S*. In addition, if l[f,g] = (g(z)/ O+aRe[zg'(z)_1] o-a.
chapter |6 pages
> 0. We show that Re \jl(pi, a) 0 by using (3.6-12) and
I 1+ I y 1+ 2 Im · Im y y I 2Re y, I + I P - y I )
chapter 4|1 pages
APPLICATIONS OF SECOND-ORDER DIFFERENTIAL SUBORDINATIONS
4.1. Second-Order Linear Differential Subordinations
chapter |1 pages
If cR(a), where R(a) is given by (4.5-10), then
Jt-l/2(1- 5/4. In the special case c = 3/2, iz
chapter |4 pages
< b < c and a [-2, 0). By employing (1.2-14) and
z U. Using this result in the above equation we obtain F'(a,b,c;z) 0 for z U.
chapter 5|7 pages
SPECIAL DIFFERENTIAL SUBORDINATIONS
5.1. Conditions for Special Subclasses of Starlike Functions
chapter 5|4 pages
3. On a Theorem of Robertson
zj"(z)/f'(z) and zj'(z)/f(z) play a fundamental role in the z/'(z) 2 S*(a), If = f(z)/[zf'(z)], fi[l,l]
chapter 5|3 pages
5. Functions with Bounded Turning And Starlike Functions
I arg f'(z) I rt/2, and arg f'(z) is the angle of turn (or rotation) of the z under the mapping f . It is well known that f S *. This section describes some of
chapter |3 pages
F: e and let 0 be a complete
)/dz] 0, and I F(wz I 1, for w e U. Since F(O) = [I lim(l-r)-
chapter |5 pages
A e and > 0 is chosen sufficiently small so that
zpoint of maximum the = mf(zo), = 1/ Df(z A 0.
chapter 7|1 pages
.3. Dominants and Admissible Functions in
f and g be f is said to be subordinate tog, written f g or /(z) g(z), if there exists a mapping ..J./(B), with = f(B) f g if and only if f(O) [50]
chapter |2 pages
If =
= {z >a}. z e -1a, then by the normalization (8.3-1) we have z + a, where e = 0,
