PAPER ( D ) *
lOn a c l a s s of mean p - v a l e n t f t m c t i o n s i n t h e u n i t c i r c l e .
.1,
Let
" '
w(z) = 1 + a
z+a^z
1
2
2
+...
be an analytic function, regular and schlicht in the unit circle
Iz^ < 1* We denote its maximum modulus for Izi = r by M(r,w),
0 < r < 1.
We shall now consider a class ¥ of analytic functions
P
P
V :
w (z) = z w(z) ,
p > 1/2, w ?^ 0 ,
P
P
'
' p
'
which a r e mean p - v a i e n t i n Iz! = r < 1 [ l , 1 G 5 ] . I f we d e n o t e t h e
mean v a l u e
1
Kr.w ) = - — X
l o g Iw^(z) i d a r g w ( z ) ,
P
2pir | 2 | = 1
• P
p
t h e n t h e r e h o l d s t h e f o l l o w i n g theorem which i s s i g n i f i c a n t
in
g i v i n g a c±ose l i n k between t h e maxirnxM siodiilus M(r,v7), which i s
u s u a l l y e a s y t o d e a l w i t h , and t h e mean I ( r , w ) , which i s
frequently
less
tractable.
Iheorem 1. Let w ( z ) € V
inO< l z l = r < 1 .
Then t h e r e
-^^
'—' P
• P —
-~~
holds the inenuality
,
r(1-r) ,
(1)
I(r,w ) < A Uog M(r,w) + p log
| ,
P
^
1+r
^rhQ^e. A. is a positive constant depending on, r. Sign of equality
holds in (1> for the function
t
p
K
~2p
w*(z) = z^ (1- e z)
€ ¥ , < real.
P
P
-Proof. Let us consider the function
,
'
i < 2p
(S)
¥ ( z ) = w Cz)/w*(z) = HU e z ) w ( z ) ,
'_P
P
where l o g 1#(0) = 0. We "assume t h a t w (z) i s r e g u l a r i n Iz ! < 1 and
P
does n o t have z e r o s t h e r e .
I f we p u t
'w^(z) = te^'^ ,
w*(z) = t^e^*^* ,
¥ ( z ) = Te^^ ,
P
p
* To be published in Vijnan Parishad Patrika^ OtK. S« ^aa**^•
37.
then in view of (2)
log T + if
log t - log t* + ±i<f^<f*)
=
,
whence equating real and imaginary parts on both sides we find that
(3)
log T = log (t/t*)
,
$ = f - «y*'.
Hov we c o n s i d e r t h e i n t e g r a l
•
1
'
1
2Tr
which on acconnt of t h e r e l a t i o n s (5) becomes
1 2ir
I ( r , w ) = " 5 - : I (log T + log t*) ( d $ + d f*)
, .
p
'^^ 0
1
2Tr
; Sir
2tr
= "5:r i" l o g T d $ + I l o g T d <?* + I l o g t*d«?+ '
^P^ 0
0
.
0
2r
+ I log t*df*
0
= Iw, + I + I,- + I •
1
2
3
4 •
How p u t t i n g l o g W(z) = l o g T + i $ , = u + i v , we g e t :
.1
= ir-- H
f 3
]—
9v
a . dv
,
(u -Tz) - "ri (u r—,) f
dx dy
,
which on application of the Cauchy-Riemann equations becomes
I- = - — J J
U -^^ ) + ( T ^ » f dx dy
Similarly we can show that
Further,
1
^o''='¥^
2 oil '^^^
I^ ^
>
0 .
Q •
I becomes in view of the relations (3)
2
2w
^ ^l°g T d<y ' - l o g T d f ) ,
'
0
and s i n c e j l o g T d $ > 0 by t h e same s y l l o g i s m a s for I , , we have
0 > 1 2B""
^
2
2pTr 0
•*
In the same way I- becomes in" view of (3)
^
i
^2w
I„ = P 7 ^ I
3
<iP^ 0
( l o g t*dcp - l o g t * d f *)
J
.
J
where by a p p l y i n g t h e same method a s for I
,
so. t h a t
1 2w
I„ <
I l o g t*d<f
.
3
2p5r 0
^
'
Thus combining a l l t h e s e r e s u l t s we o b t a i n
,
2Tr
we h a v e : I l o g t * d ^ * > 0,
,0
28
Off
(4)
I ( r , w ) < A —L. I ( l o g T dcp + l o g t*dcf )
P
2ptr 0
^
*=
3
J
( l o g IM(z) ( + l o g \z^ ( 1 - e
2pir
P
= A
z)
I>
izl=1
-d a r g w (z)
P
IM(z)llzl
log ( l - i 2 | 2 ) P
( 1 - r ) ^ P lw(z)l r P
= A log
(1-r^)P
< A l o g | ( — ) ^ rP M(r)l
»
where frora (2) we have l¥(z) 1 = ( 1 - r ) ^ P | w ( z ) I, so t h a t (1) f o l l o w s
directly from (4),
We generalise the class Y
2.
I
as followsR»106]: Let p.k be
P
positive integers, 1 < k < 4p, p > 1/2, and let
w
(z) = z^ (1 + a, '2^ + a„ z^^ + ... ) € V
P,k
1
2
p
be mean p - v a l e n t i n Iz i < 1.
k >
z = s , we g e t t h e c l a s s of
lA
Tlien by means of t h e
transformation
fiihctions
2
DA
\^hich a r e mean ( p A ) - v a l e n t i n 0 < I ^ I < 1 .
We d e f i n e t h e mean v a l u e
I(r,w
) = -=- S
l o g Iw , ( z ) l d a r g w . (z) ,
' P,k
2pir (21=1
Pjk
P>^
Theorem 2»
Let w
P>^
(5).
(2)€¥
inlz'<1.
Then we have
(pA) """
I(i',w ^) < A jlog M(r,w) + ^ l o g {r^(1-r:)/(1+r ))j
where M(r,w) = max
,
lw(^ ) I , w(C ) = 1+a.C "•• a ^ "•"••• r a^id A i^
I (J^ l=r
I
2
a pos;ltive constant depending o n r only*
'
P r o o f o f this theorem follows in t h e same manner a s i n t h e
above theorem.
¥e remark that the inequalities (1) and (5) do not,however,
give the best poasible results, but these ^bounds are interesting
on account of the elementary method of proof used here. The best
29
p o s s i b l e bounds can be evalTiated e . g . hj highly developed v a r i a ' t i o n a l methods.
The author takes t h i s opportunity of extending h i s c o r d i a l
thanks to PrGf.W.K.Hayman for various suggestion on t h i s problem.
Reference.
{jl. W.K.Hayman: The coefficients of schlicht and allied functions.
Proc. International Congress of Maths. ,Msterdam,
vol. Ill (1954),102-8.