奈良教育大学学術リポジトリNEAR
Meromorphic functions convex of order at most ρ in one direction
著者 SAKAGUCHI Koichi
journal or
publication title
奈良学芸大学紀要. 自然科学
volume 10
number 2
page range 109‑112
year 1962‑03‑26
URL http://hdl.handle.net/10105/4774
Journ. Nara Gakugei Univ., Nat. Sci., Vol. 10, No. 2, Mar., 1962 109
Meromorphic functions convex of order at most p in one direction
Koiclii SAKAGUCHI
(Department of Mathematics, Nara Gakugei University) (Received October 20, 1961)
1. Introduction.
In this paper it is our purpose to study odd functions and even ones convex of order at most p in one direction and to obtain a criterion for the multivalency of general functions not necessarily odd or even.
It has been shown by Ozaki (_1~) and Umezawa C2D that if/(z)=2H- is meromorphic in |z|O and satisfies there one of the following conditions:
® »«-m<i
1+3? /(z) <2,
zf'(z -)
/'O) <2, then f(z^) is regular, univalent and convex in one direction for Z<C|»i.
In case /(z) is odd, these conditions may be replaced by the following ones respectively.
®
1+ai fCzy f/r^>~l> 1+31 z/'O )
1+9? zf'Cz) <3,
/'GO
<3, (Corollary 1).
2. Odd or even functions convex of order at most p in one direction.
LEMMA.Let an odd or even function f(z) be meromorphic in \z\^r, and let f(z)
^fO, oo on \z\--r. Let n(0) and n(oo) be the numbers of zeros and poles in [z|<r of /(z) respectively. If a straight line XOY through the origin is cut 2p times by the image curve of \z\=r under f^z) as z traverses once on \z\=r, then for a non- negative integer k we have
(1) />=|<0)-«(oo)|+2A.
110 Kfiichi Sakaguchi
PROOF. We denote by C the above image curve. If the rays OX and OY are cut s and t times respectively by C, then we have
s=K0)-«(oo)[+2fe, *=|m(0)-«O°)|+2/\
where h, j are suitable non-negative integers. Hence 2/>=s+*=2|w(0)-M(oo)|+2(fc.+y).
Furthermore we easily find that (a) iff(z) is odd, then h=j, since C is symmetrical with respect to the origin, and (b) iff(z) is even, then both h andj are even, since C is a double curve. Hence (1) follows.
THEOREM 1. Let an odd or even functionf(z) be meromorphic in \z\^r, and let
f'(.z)-%Q, oo on \z\-r. Let n(0) and «(ck>) be the numbers of zeros and poles ofzf'(z) in \z\<r respectively. If/(z) satisfies the condition
(2) 1+3? /'GO d6<2xUn(0)-n(°°)\+2(k+l)} , z=rew,
for a non-negative integer k, then f(z) is convex of order at most |»(0)-w(»°)|+2&
in one direction for \z\^r, and it is at most iV(°°)+(«(O)-w(c»)[+2& valent there, where i\T(°<0 denotes the number of poles of/(z) in z<r.
PROOF. It is sufficient to show that zf(z~) is (starlike) of order at most \n(0)-rc(oo)|
+2k in the direction of one straight line through the origin. Suppose that the image curve C of \z\=r under zf'(z) cuts every straight line through the origin more than 2(!«(0)-n (.°°)\+2k) times. Then from the lemma, C cuts such every straight line at least 2(|w(0)- mC°°)I+2(^+1)) times, and hence we have
3f zCz/O))'
*/U) de>2n{\n(0)-n(»=)\+2(k+l)} , z=re'e.
which contradicts (2). Consequently there exists at least one straight line which is cut by C at most 2([«(0)-«C°°)I+2A) times.
Thus the theorem is proved.
THEOREM 2. Let m be a positive number such that 2(m-k-l)+n(0)-n(oo)-\n (0)-m(oc)|^o. In Theorem 1 , iff(z) satisfies the condition
, . _ ^1,»)#'M ^{"(n)-"(°')tK°)-"(°')!+2(*+i))
^
6J m<^L^-Jl ^/(z)<2(»m-*-1)+«(0)-«(oo)-|m(0)-«(oo)| ' =r,
instead of (2), £&ew we fea^e a/so ifee same conclusion for /(z).
This can be proved in the same way as used by Ozaki C10 and Umezawa C2D, and so the proof may be omitted here.
THEOREM 3. Let an odd or even function /(z) wn£fe the expansion
00
f(z)=zQ +Sa o+2n 2<2+"M C<?: positive or negative integer)
n=l
at the origin be meromorphic in \z\<r. Let p and m be a positive integer and a positive number respectively such that p-\q\ is even and non-negative, and 2m-p-\-
q-2>X). If f(z) satisfies the condition
r 4"> -m<-l+^sg-(^<rw(/'+0±^- z\<r
(. 4; m<i+Jt//O)<2»i-/>+<?-2 ' z\<-r>
Meromorphic functions convex of order at most p in one direction 111 then f(z) is convex of order at most p in one direction for \z\<r, and furthermore (a) when q>0, f(z) is regular and at most p valent in \z\<r, (b) when q<0, f(z) is at most p+\q\ valent there.
PROOF. From (4) we see that zf(z) has no zeros and no poles in 0<|;j|<r. Therefore (a) when <?>0, for every p less than and sufficiently near to r, f(z) satisfies on \z\=p the inequality (3) with k=(p-c/)/2. Hence from Theorem 2 f(z) is convex of order at most p in one direction and at most p valent for \z\<r because of p=\n(fi)-w(»°)i+2&. (b) when <7<0, we have a similar proof also.
COROLLARY 1. Let an odd or even function f(z) with the expansion f(z)=za +^ aq+2n z q+in (g: positive or negative integer)
at the origin be meromorphic in \z\<r. Let p a positive integer such that p-\q\ is even and non-negative. If f(z) satisfies one of the following conditions:
(5) l+^5fS>-tF •EW<r,
(6) 1+R^g<*±£±? , W<r,
(7) (8)
1+5R: <p+2, z\<r,
i-s+SR^ <p+2, \z\<r,
then f(z) is convex of order at most p in one direction for \z\<r, and furthermore (a) when q>0, f(z) is regular and at most p valent in \z[<r, (b) when q<0, /(z) is at most p+\g\ valent there.
PROOF. Putting m=(p-q+2)/2, m=co, m=p+2, m=p-q+2 in (4), we have (5),
(6), (7), (8; respectively.
3. A criterion for the multivalency of general functions.
From Theorem 3 we have the following for general functions not necessarily odd or even.
THEOREM 4. Let a function f(z) with the expansion
OS
f(z)=z" +2] <z3+rc z (i'- Positive or negative integer)
at the origin be meromorphic in \z\<r, Let p and m be a positive integer and apositive number respectively such that p-\q\ is even and non-negative, and Am-p-t-q-2^0. If /(z) satisfies the condition
(9) - <l+^-l*fgH"S^ ' '*<'å
then (a) when <?>0, f(z) is regular and at most p valent in jzjO, (b) when q<S), f(z) is at most p+\q valent there.
PROOF. Suppose that /(z) has a zero or a pole of order k or -k in 0<|z|<r. Then from
the fact that 9JCl+z/"Cz)//Cz)-yz//(z)//"(z)] is bounded to at least one side, we
112 Keichi Sakaguchi
easily find k-2. Consequently if we set
^V^^ +Si
»=1then g(z) is also meromorphic in |2[<Vr and is odd or even, and there holds the equality 1+ g-'U)"n 1+ /'U'2) ~2~~7tS*)"7 '
which combined with (9) gives
By virtue of Theorem 3, this deduces that (a) if q>0, then g(z) is regular and at most p valent in \z\<.^/r, and (b) if <7<0, then g(z) is at most p+\q\ valent there.
On the other hand, it is easy to see that the number of valence off(,z) in \z\<Cr is not larger than that of g(z) in z<^/r. We thus complete the proof.
Just as in the proof of Corollary 1, we have also the following COROLLARY 2. Let a function f(z) with the expansion
f(z)=Zq +^ aq+n ZCL+n (<T- positive or negative integer^
at the origin be meromorphic in |z|O. Let p be a positive integer such that p-\q\ is even and non-negative. If f(z) satisfies one of the following conditions:
UU; 1+Jt/'(*)--f^/(z)> 4 ' \z\<r>
UU l+Jt/'(z)-y>«/(z^ 4 å 3l<r>
(12) (13)
1 +3} _±ftZf%*l ^-P+2
f
