Classes of functions defined by certain differential–integral ... - Core

Journal of Computational and Applied Mathematics 105 (1999) 245–255

Classes of functions deÿned by certain dierential–integral operators J. Dziok ∗ Institute of Mathematics, Pedagogical University of RzeszÃow, ul. Rejtana 16A, PL-35-310 RzeszÃow, Poland Received 1 October 1997; received in revised form 25 February 1998 Dedicated to Haakon Waadeland on the occasion of his 70th birthday

Abstract In this paper we investigate a class of p-valent analytic functions with ÿxed argument of coecient, which is deÿned in terms of certain dierential–integral operators. Coecient estimates, distortion theorems, extreme points, the radii of c 1999 Elsevier Science Ltd. All rights reserved. convexity and starlikeness in this class are given. MSC: 30C45; 26A33

1. Introduction Let A(p; k), where p; k ∈ N = {1; 2; : : :}; p ¡ k, denote the class of functions f of the form p

f(z) = z +

∞ X

an z n ;

(1.1)

n=k

which are analytic in U = U(1), where U(r) = {z : |z| ¡ r}. Let A = A(1; 2). We say that a function f ∈ A(p; p + 1) is convex in U(r); r ∈ (0; 1]; if 

Re 1 +



zf00 (z) ¿ 0; z ∈ U(r) f0 (z)

(1.2)

and we say that a function f ∈ A(p; p + 1) is starlike in U(r); r ∈ (0; 1]; if 



zf0 (z) Re ¿ 0; f(z) ∗

z ∈ U(r):

E-mail: [email protected].

c 1999 Elsevier Science B.V. All rights reserved. 0377-0427/99/$ - see front matter PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 1 4 - X

(1.3)

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J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255

By Cp we denote the class of all functions convex in U and by Sp∗ we denote the class of all starlike functions in U. We say that a function f ∈ A is subordinate to a function F ∈ A, and write f ≺ F, if and only if there exists a function ! ∈ A; !(0) = 0; |!(z)| ¡ 1; z ∈ U, such that f(z) = F(!(z)); z ∈ U: Let p; k ∈ N; A; B; ; ;  ∈ R; − ¡ p ¡ k;  + ¿ − p; 06B61; −B6A ¡ B; (B 6= 1 or cos  ¡ 0) and let denote the gamma function. In [3, 4] Kim and Srivastava et al. studied the following integral operator: Deÿnition 1.1. Let f ∈ A(p; k) and let log(z − ) be real when z −  ¿ 0; z;  ∈ U: The integral operator Q  f(z) is deÿned for  ¿ 0 and the function f by Q  f(z) =

1 ()

Z z 0

1−

 z

−1

 −1 f() d:

Now, we deÿne the dierential–integral operator   f(z) for 60: Then there exists a nonnegative integer m and a real number ; −1 ¡ 60; such that  = − m: Deÿnition 1.2. Let f ∈ A(p; k) and let log(z − ) be real when z −  ¿ 0; z;  ∈ U: The integral operator   f(z) is deÿned for 60 ( = − m; m ∈ N ∪ {0}; −1 ¡ 60) and for the function f by   f(z)

=

d m+1 1 (1 + ) d z m+1

Z

0

z

(z − )  −1 f() d:

The multiplicities of (z − )−1 ; (z − ) in Deÿnitions 1.1 and 1.2 are removed by requiring log(z − ) ∈ R when z −  ¿ 0: By using these deÿnitions we deÿne the linear operator

 : A(p; k) → A(p; k) by

 f(z) =

(p +  + ) −  z Q f(z) (p + )

 f(z) =

(p +  + ) 1−−  z  f(z) (p + )

for  ¿ 0

and for 60:

Putting = 1 in Deÿnitions 1.1 and 1.2 we obtain the integral operator deÿned by Owa [5] and investigated by Srivastava and Owa [7], Dziok [1, 2] and others. Deÿnition 1.3. Let T (; ) denote the class of functions f ∈ A(p; k) satisfying the following condition:

 f(z) 1 + Az : (1.4) ≺ p z 1 + Bz

J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255

247

Deÿnition 1.4. Let T (; ) denote the subclass of the class T (; ) of functions f of the form (1.1), such that arg an =  for an 6= 0; n = k; k + 1; k + 2; : : : . We can write every function f from the class T (; ) in the form p

f(z) = z + e

i

∞ X

|an |z n ; z ∈ U:

(1.5)

n=k

In the present paper we obtain coecient estimates, distortion theorems, extreme points, the radii of convexity and starlikeness for the class T (; ) and coecient estimates for the class T (; ). 2. Coecient estimates By Deÿnition 1.4 we obtain: ˜ Lemma 2.1. If A6A˜ and B6B; then the class T (; ) for parameters A; B is included in the class ˜ ˜ T (; ) for parameters A; B: Lemma 2.2 (Ratti [6]). Let f be a function of the form (1.1). If f ≺ g and g is convex function, then |an |61; n = k; k + 1; k + 2; : : : . After some calculations we obtain: Lemma 2.3. If a function f of the form (1.1) belongs to the class A(p; k); then

 f(z) = z p +

∞ (n + ) (p +  + ) X an z n ; z ∈ U: (p + ) n=k (n +  + )

Theorem 2.1. If a function f of the form (1.5) belongs to the class T (; ); then ∞ X n=k

−1 n |an |6(; A; B);

(2.1)

where B−A (; A; B) = p 2 1 − B sin2  − B cos 

(2.2)

(n +  + ) (p + ) : (n + ) (p +  + )

(2.3)

and n

=

Proof. Let f ∈ T (; ). By Deÿnition 1.4 we obtain

 f(z) 1 + A!(z) = ; zp 1 + B!(z)

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J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255

where !(z) is an analytic function in U, such that !(0) = 0 and |!(z)| ¡ 1 for z ∈ U. Thus, we have  f(z) − z p = |!(z)| ¡ 1: B  f(z) − Az p Using Lemma 2.3 we obtain ∞ X n=k

where

n









−1 n−p ¡ B n |an |z

− A + Be

i

∞ X n=k



−1 n−p ; n |an |z



is deÿned by (2.3). Thus, putting z = r; 0 ¡ r ¡ 1; we have

|w| ¡ |B − A + Bwei |;

(2.4)

where w=

∞ X n=k

−1 n−p : n |an |r

Since w ∈ R, by (2.4) we have (1 − B2 )w2 − (2B(B − A)cos )w − (B − A)2 ¡ 0: Solving this inequality with respect to w we obtain ∞ X n=k

−1 n−p n |an |r

where (; A; B);

n

¡ (; A; B);

are deÿned by (2.2) and (2.3). Thus letting r → 1− we obtain (2.1).

Theorem 2.2. If a function f of the form (1.1) belongs to the class T (; ); then |an |6(B − A) where

n

n;

n = k; k + 1; k + 2; : : : ;

(2.5)

is deÿned by (2.3). The result is sharp.

Proof. Let a function f of the form (1.1) belong to the class T (; ) and let us put z : g(z) = (z −p  f(z) − 1)(A − B)−1 ; h(z) = 1 + Bz By (1.4) we have g ≺ h. Since the function g is the function of the form g(z) =

∞ X

[(A − B)

n]

−1

an z n−p

n=k

and the function h is convex in U; by Lemma 1.2 we obtain [(A − B)

n]

−1

|an |61;

n = k; k + 1; k + 2; : : : :

Thus we have (2.5). Equality in (2.6) holds for the functions gn of the form gn (z) = h(z n−p ) = z n−p +

∞ X i=n−p+1

(−B) i−n+p z i ; n = k; k + 1; k + 2; : : : :

(2.6)

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249

Thus equality in (2.5) holds for the functions fn of the form p

fn (z) = z + (A − B)

nz

n

+

∞ X

(A − B)

i

(−B) i−n z i ; n = k; k + 1; : : : :

i=n+1

By Theorem 2.1 we obtain: Corollary 2.1. If a function f of the form (1.5) belongs to the class T (; ); then |an |6(; A; B) where (; A; B);

n;

n = k; k + 1; k + 2; : : : ;

are deÿned by (2.2) and (2.3).

n

Putting  =  in Corollary 2.1 we obtain: Corollary 2.2. If a function f of the form (1.5) belongs to the class T (; ); then |an |6 where form

n

B−A 1+B

n;

n = k; k + 1; k + 2; : : : ;

is deÿned by (2.3). The result is sharp. The extremal functions are functions fn of the

fn (z) = z p −

B−A 1+B

nz

n

; n = k; k + 1; k + 2; : : : :

(2.7)

3. Distortion theorems and extreme points Theorem 3.1. If f ∈ T (; ); |z| = r ¡ 1; then for 60 r p − (; A; B)

k

r k 6|f(z)|6r p + (; A; B)

k

rk

(3.1)

and for 6 − 1 pr p−1 − k(; A; B) where (; A; B);

n

kr

k−1

6|f0 (z)|6pr p−1 + k(; A; B)

k−1

;

(3.2)

are deÿned by (2.2) and (2.3).

Proof. Let f ∈ T (; ); |z| = r ¡ 1: Since the sequences { decreasing and positive, by Theorem 2.1 we obtain ∞ X

kr

|an |6(; A; B)

for 60;

k

n}

for 60 and {

n =n}

for 6 − 1 are (3.3)

n=k ∞ X n=k

n|an |6k(; A; B)

k

for 6 − 1:

(3.4)

250

Since

J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255

∞ ∞ X X p i n |f(z)| = z + e |an |z 6r p + |an |r n n=k

= rp + rk

∞ X

n=k

|an |r n−k 6r p + r k

n=k

and

∞ X

|an |

n=k

∞ ∞ X X p i n p |f(z)| = z + e |an |z ¿r − |an |r n n=k

p

=r −r

k

∞ X

n=k

|an |r

n−k

p

¿r − r

k

n=k

∞ X

|an |;

n=k

by (3.3) we obtain (3.1). Using (3.4) the estimations (3.2) we prove analogously. Putting  =  in Theorem 3.1 we obtain: Corollary 3.1. If f ∈ T (; ); |z| = r ¡ 1; then for 60 B−A B−A k p k rp − k r 6|f(z)|6r + kr 1+B 1+B and for 6 − 1 B−A B−A k−1 k−1 pr p−1 − k 6|f0 (z)|6pr p−1 + k ; kr kr 1+B 1+B where n is deÿned by (2.3). The result is sharp. The extremal function is function fk of the form (2.7). Theorem 3.2. Let fk−1 (z) = z p and let fn ; n = k; k + 1; k + 2; : : : ; be deÿned by (2.7). A function f belongs to the class T (; ) if and only if it is of the form f(z) =

∞ X

n fn (z);

z ∈ U;

(3.5)

n=k−1

where

∞ X

n = 1 and n ¿0; n = k − 1; k; k + 1; : : : :

n=k−1

Proof. (⇒) Let a function f of the form f(z) = z p −

∞ X

|an |z n

n=k

belong to the class T (; ): Let us put 1 + B −1

n = |an |; n = k; k + 1; k + 2; : : : B−A n

(3.6)

J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255

251

and ∞ X

k−1 = 1 −

n :

n=k

We have n ¿0; n = k; k + 1; k + 2; : : : and by (1:7) we have k−1 ¿0: Thus, ∞ X

n fn (z) = k−1 fk−1 (z) +

n=k−1

∞ X

n fn (z)

n=k

= 1−

∞ X

! p

n z +

∞ X 1+B

n=k

=z p −

∞ X 1+B

B−A

n=k

−

n=k

∞ X

−1 n |an |

B−A

−1 p n |an |z

+



∞ X 1+B n=k

B−A

B−A z − 1+B p

nz

n



−1 p n |an |z

|an |z n = f(z)

n=k

and the condition (3.5) follows. (⇐=) Let a function f satisfy (3.5). Thus, ∞ X

f(z) =

n fn (z) = k−1 fk−1 +

n=k−1

n fn (z)

n=k

= 1−

!

∞ X

n z p +

n=k

= zp −

∞ X

n=k

∞ X B−A n=k

∞  X

1+B

n n z

zp −

B−A 1+B



n

n nz

n

and we can write the function f in the form (3.6), where |an | = n

B−A 1+B

n:

Since ∞ X n=k

−1 n |an |

=

∞ X n=k

n

B−A B−A B−A = (1 − 1−k )6 ; 1+B 1+B 1+B

we have f ∈ T (; ), which ends the proof. By Theorem 3.1 we have: Corollary 3.2. The class T (; ) is convex. The extremal points are functions fn of the form (2.7) and the function fk−1 (z) = z p .

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J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255

4. The radii of convexity and starlikeness Theorem 4.1. If f ∈ T (; ); then a function f is convex in the disc U(rc ); where rc = inf

n¿k

n2 (; A; B) 2 p

and (; A; B);

n

!1=(p−n)

(4.1)

n

are deÿned by (2.2) and (2.3).

Proof. Let a function f of the form (1.5) belong to the class T (; ). By (1.2) the function f is convex in U(r); 0 ¡ r61; if 00 1 + zf (z) − p ¡ p; z ∈ U(r): (4.2) 0 f (z) Since i P∞ 2 00 n−1 e (z) (n − np)|a |z zf n n=k 1 + P − p = p−1 n−1 pz f0 (z) + ei ∞ n|a |z n n=k P ∞ (n2 − np)|a k z|n−p n n=k P 6 ; n−p p− ∞ n=k n|an k z|

putting |z| = r, condition (4.2) is true if ∞ X n2 n=k

p2

|an |r n−p 61:

(4.3)

From Theorem 2.1 we have ∞ X

[(; A; B)

−1 n ] |an |61;

(4.4)

n=k

where (; A; B);

n

are deÿned by (2.2) and (2.3). Thus it is sucient to show that

2

n n−p r 6[(; A; B) p2

n]

−1

for n = k; k + 1; k + 2; : : : ;

that is r6(bn )1=(p−n)

for n = k; k + 1; k + 2; : : : ;

(4.5)

where

n2 n: p2 Condition (4.5) is true for r = rc , where rc is deÿned by (4.1). Since bn = (; A; B)

(4.6)

bn ¿ 0; n = k; k + 1; k + 2; : : : and lim (bn )1=(p−n) = 1;

n→∞

we have 0 ¡ rc 61; which ends the proof.

(4.7)

J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255

253

By Theorem 4.1 we have: Corollary 4.1. Let 6 − 2 and let (; A; B); 2

k p2

n

be deÿned by (2.2) and (2.3). If

k (; A; B)61;

(4.8)

then T (; ) ⊂ Cp :

(4.9)

In the other case a function f ∈ T (; ) is convex in U(rc ); where k2 rc = (; A; B) 2 p

!1=(p−k)

:

k

(4.10)

Proof. Let f ∈ T (; ). By Theorem 4.1 the function f is convex in U(rc ), where rc is deÿned by (4.1). For 6 − 2 the sequence {bn }, deÿned by (4.6), is decreasing. Hence, if bk 61; that is condition (4.8) is true, then (bn )1=(p−n) ¿1 for = k; k + 1; k + 2; : : : : This, by (4.7) we have rc = 1; that is condition (4.9) is true. In the other case we have (bk )1=(p−k) 6(bn )1=(p−n) for n = k; k + 1; k + 2; : : : ; which gives (4.10). Since for 6 − 2 k2 (; A; B) 2 k ¡ 1; p by Corollary 4.1 we have: Corollary 4.2. If 6 − 2; then T (; ) ⊂ Cp : We now determine the radius of starlikeness for the class T (; ). Theorem 4.2. If f ∈ T (; ); then the function f is starlike in U(r ∗ ); where 

n¿k

and (; A; B);

n



1=(p−n) n : n p are deÿned by (2.2) and (2.3).

r ∗ = inf (; A; B)

(4.11)

Proof. Let a function f of the form (1.5) belong to the class T (; ). By (1.3) the function f is starlike in U(r); 0 ¡ r61; if 0 zf (z) ¡ p; z ∈ U(r): (4.12) − p f(z)

254

Since

J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255

0 i P∞ n zf (z) e (n − p)|a |z n n=k f(z) − p = z p + ei P∞ |a |z n−1 n n=k P∞ n=k (n − p)|an ||z|n−p ; P 6 1 − ∞ |a ||z|n−p n

n=k

putting |z| = r, the condition (4.12) is true if ∞ X n n=k

p

|an |r n−p 61:

(4.13)

By (4.4) it is sucient to prove n n−p r 6[(; A; B) n ]−1 for n = k; k + 1; k + 2; : : : ; p that is r6(hn )1=(p−n) ; where

n n: p The last inequality is true for r = r ∗ , where r ∗ is deÿned by (4.11). Since hn = (; A; B)

hn ¿ 0 for n = k; k + 1; k + 2; : : : and lim (hn )1=(p−n) = 1;

n→∞

we have 0 ¡ r ∗ 61; which completes the proof. From Theorem 4.2 we obtain the following corollaries analogously as Corollaries 4.1, 4.2 from Theorem 4.1. Corollary 4.3. Let 6 − 1. If k (; A; B) k 61; p where (; A; B); n are deÿned by (2.2) and (2.3), then T (; ) ⊂ Sp∗ : In the other case a function f ∈ T (; ) is starlike in U(r ∗ ); where 

k r = (; A; B) p ∗

1=(k−p)

k

Corollary 4.4. If 6 − 1; then T (; ) ⊂ Sp∗ :

:

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255

References [1] J. Dziok, Classes of p-valent analytic functions with ÿxed argument of coecients, I, to appear. [2] J. Dziok, Classes of p-valent analytic functions with ÿxed argument of coecients, II, Proceedings of the Fifth Finish-Polish–Ukrainian Sommer School in Complex Analysis, Poland, Lublin, 1996, to appear. [3] I.B. Jung, Y.C. Kim, H.M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl. 176 (1993) 138–147. [4] Y.C. Kim, K.S. Lee, H.M. Srivastava, Certain classes of integral operators associated with the Hardy space of analytic functions, Complex Variables Theory Appl. 20 (1992) 1–12. [5] S. Owa, On the distortion theorems, Kyungpook Math. J. 18 (1978) 53–59. [6] J.S. Ratti, The radius of univalence of certain analytic functions, Math. Z. 107 (1968) 241–248. [7] H.M. Srivastava, S. Owa, An application of the fractional derivative, Math. Japon. 29 (1984) 383–389.