Meromorphic functions multivalently starlike in a wide sense

奈良教育大学学術リポジトリNEAR

Meromorphic functions multivalently starlike in a wide sense

著者 SAKAGUCHI Koichi, WATANABE Shigeyoshi journal or

publication title

奈良学芸大学紀要. 自然科学

volume 11

page range 3‑7

year 1963‑02‑28

URL http://hdl.handle.net/10105/3475

Journ. Nara Gakugei Univ., Nat. Sci., Vol. ll, Mar. 1963

Meromorphic functions multivalently starlike in a wide ide sense

Kolchi SAKAGUCHI and Shigeyoshi WATANABE

(Department of Mathematics, Nara Gakugei University) (Kojima High School)

(Received Sept. 30, 1962)

1. Introduction

Let /(z) whose expansion at the origin is z" + be meromorphic in |z[<l and have there exactly p zeros and exactly q poles, where multiple zeros and poles are counted in accordance with their multiplicities and p^O, q^O, p^-q.

(1) If there exists a positive number R such that either

(1.1) 9W0O//(z)D>0, z=r,

holds for every r in R<Cr<l together with p>q, or

(1.2) SC^(2)//(O)<0- z=r,

holds for every r in J?<r<l together with p<.q,

then /(z) is said to be multivalently starlike of order (p,q,s) with respect to the origin for |zj<l, and the class of such functions is denoted by S(p,q,s).

(2) If for an arbitrary positive number e, there exists another positive number R(s*) such that either

dargf(re«)>-e , 0!<02 )

91

holds for every r in R<Cr<Ll together with fi~>q, or

( 1.4) \ daigKrem)<e, e1<03)

Jei

holds for every r in i?<r<l together with p<Cq,

then /(z) is said to be multivalently starlike in a wide sense of order (p,q,s) with respect to the origin for |z|<l, and the class of such functions is denoted by S*(p,q,s).

From the definitions, it is evident that S*(p,q,s)r>S(p,q,s) and every function of S(p,q,s) is maxCi>,<jO-valent in |z|<l.

Recently Bender (V) studied the class of functions /(z) given by the representation

(1.5) /(2) =C«<2)>2S TKl-^.Xl- ^O, p>s>o,

Kfiichi Sakagdchi and Shigeyoshi Watanabb

where ^(z)eS(l,O,1), 0<|«j|<1, j=l,2,---, p-s. He there proved that this class of functions properly contains S(p,0,s) if ^>>s>0, and moreover that every function given by (1.5) is ^-valent in \z\<l.

In this note we shall show that this class of functions studied by Bender is equivalent to our class S*(p,0,s), ^>3gs>0. More generally, we shall give a representation theorem for functions belonging to the general class S*(i>,<7,s) and prove that S*(p, Q,s) properly contains S(p,g,sy if p-maxCs,CT]>0 or (7+minCs,CT]>0, and moreover that every function of S*(p,q,s) is maxCj>,q~)-valent in lz|<l.

2. The representation theorem for functions belonging to

S%p,q,s).

LEMMA. Let f(z)=aiz+ , ai^fO, be regular in |z|^l and have there no zeros

except at the origin. Iff(z) satisfies the condition

("2

J«l daTgf(e*)>-K (K>0) , 6i<82 ,

then there exists a function <p(z~)=ciz+ starlike with respect to the origin for z[<l such that

arg /O)

<ptz)

<£, |z|<l, and |cx|=l.

This can be proved by the same argument as used by Kaplan (2~), and so the proof will be omitted.

THEOREM 1. A necessary and sufficient condition for f(z) to be a member of the

class S*{p,q,s') is that /(«) has a representation of the form (2.1)

where ip(z)eS(l,O,l) and

/ (zX^(z)T å !i4(z)B(z), p*Q,

A(z)=n(l-^Xl-a;jz), /=/>-maxCs,O:, 0<|«j|<l, ;=1,2,-, /.

- -1

£(z)= n C(l-|-Xl-fez)3. m=q+mmCs,02, 0<|&|<l, #=1,2,-, W.

fc-i ^fc

REMARK 1. From the definitions of I and m, it holds that

m-l=s-p+q.

PROOF. (1) Proof for the sufficiency. We suppose that /(z) has the representation (2. 1).

Then/(z) is meromorphic in |z|<l and has an expansion of the form/(2)=zs +•E•E•E,and evidently it has exactly p zeros and exactly q poles in \z\<\. Moreover we see that for an arbitrary positive number e , /(z) satisfies either

S ^{92 91} dargKrea)<s, 6i<62,

Jf(l

accordig to p>q or p<g for every r«l) sufficiently near to 1, since

å «2

(2.2) 1

daigf(reie)>-e, 0i<03, or

>r p<Cq for every r(<l) suffi

S ^{91 fl2 å >0 for r-»l.}

Meromorphic functions multivalently starlike in a wide sense

Hence /(z) is a member of S*(p,q,s^).

(2) Proof for the necessity. We suppose that f(z')£S*-(p,q,s') and the zeros, the poles of/(z) in 0<[z|<l are aj, /=1,2,•E•E•E,/, and 8k, k=l,2,---, m, respectively, where necessarily l=p-maxCs.CO, 7M=<?+mmCs ,C0 hold.

Set

g<Lz~) = Cf(z^z' -m/ACz)B(z}31/(j>-q) = z+ - ,

where j4(z),fi(z) are defined above, then from (2.2) and /(2)eS*(/i,g,s), we see that

gQz) is a member of the class S*(l,0,l). Therefore, for a sequence of positive numbers

\£n} converging to zero, there exists another sequence {rn} such that rn <1, limTn=l and darg^(r,,e'«)>-£« , ei<e2.

The function gQrnZ^) satisfies the hypothesis of the lemma with K-ea , so that we can

find a function ^>m(z)=cMiZ+ starlike with respect to the origin for jz|<l such that

(2.3) aTRgO nz2 <f ^.e» , |z|<l, and ]cBl|=l.

Since the functions <?k(z), n=l,2, , form a normal family in |z(<l, there exists a subsequence {<pnmQz)} which converges uniformly in every closed subdomain interior to

|z|<l. Let n in (2.3) take values of the sequence {fc} and let m^>°°, then we have

arg ^{gCz) =0, z<l,}

where (pQz) is the limit of the sequence {<pnm(iz)}. Hence we have

(2. 4) Jgr(2)/^(z)^c(positive constant) ,

«?(z) being a function starlike with respect to the origin for \z\<X such that \<p(z)/z\=l for

2=0. Considering the value of gQz~)/ip(z) at z=0, we find from (2.4) that c=l and

<p(z)/z=l for 2=0. Thus we have

fi-(2)=K2)eS(l,0,l), whence (2.1) follows. The proof is therefore completed.

This theorem shows that the class of functions given by (1.5) is equivalent to S*(£,0,s).

P^s>0.

3. The relation between the classes S*(p,qr;s) and S(p,q,s).

THEOREM 2. The classes S*(p,q,s~) and S(p,q,s^) are equivalent, ifp-maxCs,0J=q+

minCs,03=0. The class S*(p,q,s) properly contains the class S(p,q,s), if p-max Cs,0^]>0 or (7+minCs,0^1>0. Moreover, every function of S*(p,q,s~) is maxC^,<j0- valent in |z[<l.

PROOF. (1) The case where p-maxCs,0)=Q+minCs,0D=0. In this case we have either

s=p, q-Q or p=0, s=-q. Therefore from Theorem 1 it follows that S*(P,q,s)=S(p, g,s), since

<p0OeS(l,0,l) ~^ t<p(z)y'-«eS(p-q,0,p-q) or S(0, \p-q\, p-q)

according to p>q or p<^q.

Koichi Sakaguchi and Shigeyoshi Watanabe

(2) The case where p-maxCs,C0>0. With l=p-maxCs.CO, w2=<7+minCs,(O, we set (3.1) F(z) = l a-*)2; z-1 ci+fxn-**) ; 2á", a+fxi+Bz) ej

where 0<«<l, 0<|3<l. Then F{z) is a member oiS*(p,q,s) from Theorem 1, and satisfies the equality

with C=a+l/<*(>2), D=0+l/8(>2).

Let M(z)=9fCzF'(z)/F(2)] , «(2)=3CzF'C2)/FCz)D, then w(ei«)=0 and

\ rti'a^=d'>^ie'>--iP-^4.oi 2+CcosS _ o 2+DcosO v K°J- dd l-cos^ "^ *(C+2cos/?)2 m(D+2cos6y

For 0i,02 such that cosfli=(C2-8)/2C and cos63=-1, we have

V'CB >--2g(/>-«7).i ^ C2 C(4C+C^£>-8) 8+2C-C2 ' 4C2-4 "*(OD+C2-8)2

wen^= -&-«>.

' v"aJ "

2/ , 2m

C-2 'D-2. iTj

Therefore we can find C and D such that F/(e1)>0, F>(02)<O, C>2, D>2. (Consider a value of C larger than and sufficiently near to 2). Consequently, if we denote such values

of C, D by C0=a0+l/a0, D0=8B+l/Bo respectively, then for C=C0, D=D0, u(z) takes

negative values at some points in r<J2|<l and takes positive values at some other points in r<\z\<3- for every r in 0</"<Cl. so that F(2) is not a member of S(p,g,s} when a=ao, J3=J3o.

On the other hand, from the definitions we have S*(p,q,s)z)S(p,q,s). Consequently S*(p,q,s) properly contains S(p,q,s).

Quite similarly we can prove that S*(p,q,s) properly contains S(p,q,s~) if <7+minCs,0]

>0.

Finally, to prove that every function of S*(p,q,s) is maxC/>,<50-valent in |z|<l, for f(.z) given by (2.1) we consider the function

fn (z)=C<p(rnz)/my-« zm-'A(z)B<:z), rn=- _n+V w-

This function /n(z) is maxC^.^Il-valent in lz|<Cl, since it is multivalently starlike of order (p,q,s) with respect to the origin for |z]SJl.

Now, let E* be the domain obtained from the disc |zj<l by excepting the poles of/(z), then fnCz} is regular in E* and the sequence {fn(.z~)} , n=l,2, , converges uniformly

to /(z) in every closed subdomain interior to E*. On the other hand, each/,,(z) is of

course maxCp,q^}-valent in E'*. Consequently/(z) is maxC£,<7)-valent in E*, so thatf(z) is so also in |z|<l. Thus the theorem is proved.

REMARK 2. It has been asserted by Goodman (3^ and Bender QD that the function

F(_z^) defined for q=0, ^)>s>0 by (3.1) is amember of the class S(p,0,s). This assertion, however, is not true as shown by our proof for the case 2.

REMARK 3. A function/(z) given by (2.1) is maxtp,q^-valent in |z]<l also when

p=Q. This can be proved as follows.

Set

Meromorphic functions multivalently starlike in a wide sense

c o=/oo n c-aj) n (•E ^j =l

^{0*)= J ^} =l ^{(g-«Q (l-a,jZ)} *=i Cz~#O U-PmO- ^g Then for an arbitrary complex number w0 not real, the image curve of |z|=r(<l) under g(z) does not surround the point Wo if r is sufficiently near to 1, since g(em) is real and positive. Therefore the numbers of zeros and poles in |z|<Cl of g(z)-w0 coincides with each other, and so g(z) has exactly p w0-points in |z|<l.

Next, for a real number Wi, we consider the equation g(z~)=Wi. The degree of this

equation is 2p, and if g(g)-Wi, if|<l, then we have g(_l/i)=W1, |l/f[>l. Therefore the number of roots in |z|<l of g(z)=Wi is not larger than p. Thus we see that g(z) is p-valent in 2<1, and /(z) is also so.

References

(1) J. Bender, Some extremal theorms for multivalently star-like functions, Duke Math. J. , 29 (1962) ,101-106.

(2) W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. , 1 (1952),169-185.

(3) A.W. Goodman, On the Schwarz-Christoffel transformation and p-vedent functions, Trans.

Amer. Math. Soc, 68 (1950), 204-223.