An Extension of Typically Real Functions

ANNALES

UNIVERSITATIS MARIAE C U RI E-S K LO D 0 W S K A LUBLIN-POLONIA

VOL. XLVIII, 14________________SECTIO A_________________________1994

Jan SZYNAL (Lublin)

An Extension of Typically Real Functions

Abstract. For a fixed A>0 let 7r(A) stand for the class of functions f defined by theformula f(z)=J1^ z(l—2xz+z3)~* dit(x) , where n is a probability measure on[—1,1].

ObviouslyTr(1) coincides with theclassoftypically real functions. Some convolution and coefficient results previouslyestablished for 7^r(1) are extended to the class Th(A).

1. Introduction

Let Ai (D) denote the class of holomorphic functions (1) f(z) = z + a2z2 + ••• ,

in the unit disk D = {z : |z| < 1} .

By Tft(A), A > 0 we denote the subclassof Ai(P) consisting of functions f which have the integralrepresentation

(2)

where /z is aprobability measureon the interval [—1,1] .

If SÂ(a), —oo < a < 1 , is the family of holomorphicfunctions of the form (1) which are starlike of order a in D and have real coefficients, then we see that the function

z

(1 - 2xz + z2)A ’

(3) sÄ(z,x):= x 6 [-1,1] , z e D ,

194 J. Szynal

is in 5^(1 — A) because

Re££k(^ = 1_2A + 2ARe SA(z,l)

1 — 12 1 — 2xz + z2

> 1 - A z G D .

This fact implies that Tr(\i) C Tr(A2) for Aj < A2 . Because Tk(0) = {2} in what follow we assume A > 0 .

Let us observe that Tr(1) = Tr is the well-known class of typically-real functions [1], [6], [11]. Moreover, the class Tr(\) is a convex set in the space Ai(i?) which is a locally convex linear topological space with the respect to the topology given by uniform convergenceon compact subsets ofD. So by Krein-Milman theorem every convex functional on Tr(\) attains its extremal values on the extreme points of 7r(A) [6]. It has been proved by Hallenbeck [2]

that

(4) Th(A) = co5^(l - A) , extTR(A)= {sA(z,x): x € [-1,1]} . Thefollowing two results are known for typically-real functions:

Theorem A (Robertson [8]). If f(z) = z + ^^=2anzn € Tr

and g(z) = z + ^^=2 € Tr , then

9)(2)

n=2

Theorem B (Leeman [4]). If f € Tr , then

t n - a„ < ^n(n2 -1)(2 - a2) , n = 3,4,... .

o

Alternativeproofs of Theorem A and Theorem B were presented by Krzyz and Zlotkiewicz in [3] and by Ruschewevhin [9] and [10].

In this note we extend in an appropriate way Theorem A and Theorem B to the class T/j(A). We will use convolution results of

On An Extensionof Typically Real Functions ... 195

Ruscheweyh[9] and Lewis [5] and the properties ofGegenbauer poly­

nomials C$,X)(x) , A > 0 , x £ [—1,1] , n = 0,1,... , which are defined by the generating function

<5>

2. Statements of results

In what follow we will use the following notations:

(«)„ :=a(o + 1)...(a + n- 1) , n = 1,2,... , (a)0 = 1 , a 0 ,

(6) 1) ^Z

(1 - z)™ _n=l ^{A„(A) =}

(2A)„_i (n - 1)! •

Theorem 1. If f E 7r(A) , then (7)

Inequality (7) is sharp and the extremal function has the form (6).

Theorem 2. If f(z) = z + Y^=2a*zn € TrW and = 2 + ZZ^+2 ^n2" € T«(A) , then

(8)

Corollary 1. If A = 1 then we have Robertson’s result [8]

(Theorem A).

Corollary 2. If A = 1/2 , then we have the result that the class Tr(1/2) = cbSfl(l/2) is closed under Hadamard product.

196 J. Szynal

Theorem 3. If f E Tfl(A) , then the following sharp estimate holds

(9) (n-1)! °n- (n-2)! (2A ^,n-3,4,....

For the function f(z) = s\(z,x) we have

Um = (2*+ 2)-;

i—l- 2A — 02 (n — 2)!

Corollary 3. If f € S£(l — A) , A > 0 (9) holds.

thenthe sharp estimate

Corollary 4. If Cn(x) , n = 1,2,... , A>0, wo Gegenbauer polynomial, then

dÄ)(l)-dA)(a) < (2A + 2)n_1

CiA)(l) -CjA)(x) “ (« “ !)'• for x E [-1,1] .

3. Lemmas

For the proof of Theorem 2 we need the following two lemmas.

Lemma 1 [9]. Let V C Ai(2?) with W = cö V compact.

Assume there is a function h in Ay(D') such that for all f,g EV we have

(10) h*1/2 f *1/29 eW . Then (10) holds for all f,g EW .

Lemma 2 [5]. Let S*(q) , —oo < a < 1 denote the class of a starlike functions in D . If f(z) = z + 52^2 an^n € S*(o) , g(z) = z + 52^=2 ^n2;n e ^en

(11) (/*!-« ^?)(*) = 52 ^anbn

A„(l - a)^zn E S'(a) .

On An Extension of Typically Real Functions ... 197

Lemma 3. Let

= ^e

~\k ki , n-1,2,... , t=i

v (j -1)'

£ <2A>>

_ V' f A + fc — 1 \ (2A)fc_i A ) (k- 1)!

= ^e

( :

, j — 1,2,..., n — 1,2)..

Then the following identities hold

•Sn — ^{(2A +} l)n n!

(12) _ n \ (2A + 2)n-i - 2A“(„ - 1)!“

= n(2A+ n)

” 2A(2A +1) ’ ,2,

Proof. The proof of all identities (12) is based on induction argument. We will prove the third equality of (12). Formula (12) for rn is true for n = 1 and let us assume that it is true for (n — 1) . Then we have

(n - 1)! (n - 1)(2A +n - 1) (2A)n Kn~ 2A(2A + 1) (w ~ !)• V'/i , k ~ l\(2A)t-i

(2A)„ A \k-iy.

(n — 1)(2A + n — 1) (n — 1)!

2A(2A + 1) + (2A)„

X {1+(l + |)^r+ ••• + (! + n - 1 (2A)n_i 1 A \n-l)!j

>

n = 0,1,... , , n — 1,2,...,

198 J. Szynal

_ (n - 1)(2A + n - 1) (n -1)! f 1 1 2A(2A + 1) + (2A)n + Aan-iJ

(n — 1)(2A + n— 1) (n - 1)! f(2A)n_1 n(2A + 2)n_2) 2A(2A + 1) + (2A)n t(n-l)!+Z (n - 2)! J n(2A + n)

“ 2A(2A + 1) ’ which ends the proof.

4. Proofs oftheorems

Proof of Theorem 1. From theintegralrepresentation (2) and (5) we find that

|a„| < max |Cii\(x)| .

—1<X<1

Using the integralformula for Gegenbauer polynomials [7]

(2A)'wr(A+|) n!r(l)r(A)

X

y

we get after some manipulation with Euler Gamma function that (13) |CiA)W| < fax 61-1,1],

» which implies (7).

Proof ofTheorem 2. Let f,g€ Tr(X) . We will applyLemma 1 and 2. In our case by (2) and (4) we have

v = : sx(z’ l) = (i - aJ + ’1 € I"1’11} •

W = co V = Tr(A) .

On An Extension of Typically Real Functions ... 199

Let us put

oo '

i.(2) = £^1(A)z" , A„(l/2)= 1.

i=l

Then we have

(/ *A$)(*) = £ T7??Zn = *1/2 f*1/2 <?)(*) • n=l

If f and g are in V, then they are star like oforder (1 — A) and by Lemma 2 so does f g , which implies (h *j/2 f *1/2 g) £ W . Applying Lemma 1 we end the proof.

Proof of Theorem 3. For /(z) = z+ a^zn e ?r(A) we

define the coefficients Bn , n = 1,2,... , by the relation

nB„_i = nan+1 - 2(A +n - l)a„ + (2A + n - 2)an-i ,

(I4) i n

ai = 1 , a0 = 0 , From (2) we know that

(15') °n = / C'i-ii1)^1) , n = 1,2,... .

Using the recurrenceformula for Gegenbauerpolynomials [7]

nC<A\x) -2x(A + n - l)C<A_\(x)

(16) +(2A + n - 2)c£)2(x) = 0 , n = 2,3,... , CiA) = l ,. C<A)(x) = 2Ax , CW(1) = 2A„/n! , we find from (15') and (16) that

. nB„_! = / [nCiA\i) -2(A + n - l)C^\(i) + (2A + n- 2)C^2(i)] d/x(x)

/

= -A + "~X f -2Ai) d„(x) .

200 J. Szynal

By (13) we canwrite

(17) nB„_i = (1 + ’ n = 2, ••• ,7i = l where 7„ G [—1,1] . From (14) wefind

n

(18) kBk-i = na„+1 - (2A + n - l)a„ , n = 1,2,... .

1=1

x *

After some calculations from (18) and (17) together with Lemma 3 one can get thefollowing identity (n > 2)

an_ = i^±(O2 _ 2A) (n — 1)! (n — 1)!

1 L(2A)i

n 1! „ (n— 2)!

*?1 + z_(2A)_.x ₂

S2

Sn-1

where Sn = 52^=1(1 4- ^J~)^k-\y • Taking into account that

l7n| < 1 , n = 1,2,... , we havefrom the above relations (2A)„_i (2A)ra_i.

(„_!)! a"-(„-i)!(2A a2>n-i- Applying Lemma 3 we find (9).

REFERENCES

[1] Duren, P.L., Univalent functions, A Seriesof Comprehensive Studies in Mathematics 259, Springer Verlag 1983.

[2] Halienbeck, D.J., Convex hulls and extreme points offamilies of starlike and close-to-convex mappings, Pacific J. Math. 57 (1975), 167- 176.

[3] Krzyz, J.G. and E. Zlotkiewicz, Two remarks on typically-real functions, Ann. Univ. MariaeCurie-SklodowskaSect. A 30 (1976), 57-61.

[4] Leeman, G.B., A local estimate for typically-realfunctions, Pacific J.

Math. 52 (1974), 481-484.

On An Extension of Typically Real Functions ... 201

[5] Lewis, J.L., Convolutions ofstarlike functions, Indians Univ. Math. J.

27 (1978), 671-688.

[6] Pommerenke, Ch., Univalent functions, Vandenhoeck and Ruprecht, Gottingen 1975.

[7] R a i n v i 11 e,E., Specialfunctions, Mac Millan Company, New York 1965.

[8] Robertson, M.S., Applications of a lemma of Fejér to typically-real functions , Proc. Amer. Math. Soc. 1 (1950), 555-561.

[9] Ruscheweyh, St., Convolutions in geometric function theory, Les Presses de l’Universitéde Montréal 1982.

[10] Rusch eweyh, St., Nichtlineare Extremalprobleme für holomorphe Stieltjesintegrale, Math. Z. 142 (1975), 19-23.

[11] Schober, G., Univalent Functions- Selected Topics, Lecture Notes in Mat., No.478, Springer-Verlag 1975.

Instytut Matematyki UMCS Plac M. Curie Skłodowskiej 1 20-031 Lublin, Poland

e - mail: jsszynal@golem.umcs.lublin.pl

Druk: Zakład Poligrafii Wydawnictwa ITICS

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UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA

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