Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol105.Issue1-2.1999:
Journal of Computational and Applied Mathematics 105 (1999) 245–255
Classes of functions de ned by certain di erential–integral
operators
J. Dziok ∗
Institute of Mathematics, Pedagogical University of Rzeszow, ul. Rejtana 16A, PL-35-310 Rzeszow, 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 xed argument of coecient, which is de ned
in terms of certain di erential–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 +
zf′′ (z)
¿ 0; z ∈ U(r)
f′ (z)
(1.2)
and we say that a function f ∈ A(p; p + 1) is starlike in U(r); r ∈ (0; 1]; if
zf′ (z)
¿ 0;
Re
f(z)
∗
z ∈ U(r):
E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
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(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:
De nition 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 de ned for ¿ 0 and the function f by
Q f(z) =
1
( )
Z z
0
1−
z
−1
−1 f() d:
Now, we de ne the di erential–integral operator f(z) for 60: Then there exists a nonnegative
integer m and a real number
; −1 ¡
60; such that =
− m:
De nition 1.2. Let f ∈ A(p; k) and let log(z − ) be real when z − ¿ 0; z; ∈ U: The integral
operator f(z) is de ned 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
z
(z − )
−1 f() d:
0
The multiplicities of (z − ) −1 ; (z − )
in De nitions 1.1 and 1.2 are removed by requiring
log(z − ) ∈ R when z − ¿ 0:
By using these de nitions we de ne the linear operator
: A(p; k) → A(p; k)
by
f(z) =
(p + + ) −
z Q f(z) for ¿ 0
(p + )
f(z) =
(p + + ) 1− −
z
f(z) for 60:
(p + )
and
Putting = 1 in De nitions 1.1 and 1.2 we obtain the integral operator de ned by Owa [5] and
investigated by Srivastava and Owa [7], Dziok [1, 2] and others.
De nition 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
De nition 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 De nition 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 +
∞
(p + + ) X
(n + )
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
−1
n |an |6(; A; B);
(2.1)
n=k
where
and
B−A
(; A; B) = p
2
1 − B sin2 − B cos
n
=
(n + + ) (p + )
:
(n + ) (p + + )
Proof. Let f ∈ T ( ; ). By De nition 1.4 we obtain
f(z) 1 + A!(z)
=
;
zp
1 + B!(z)
(2.2)
(2.3)
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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
n−p
−1
¡ B
n |an |z
− A + Be
i
∞
X
n=k
n−p
−1
;
n |an |z
is de ned by (2.3). Thus, putting z = r; 0 ¡ r ¡ 1; we have
|w| ¡ |B − A + Bwei |;
(2.4)
where
w=
∞
X
n−p
−1
:
n |an |r
n=k
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−p
−1
n |an |r
¡ (; A; B);
n=k
where (; A; B);
n
are de ned 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 de ned 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)
J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255
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 = k; k + 1; k + 2; : : : ;
n;
are de ned 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 = k; k + 1; k + 2; : : : ;
n;
is de ned 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|f′ (z)|6pr p−1 + k(; A; B)
k−1
(3.2)
;
are de ned 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)
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J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255
Since
∞
∞
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
p
n
|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
6|f′ (z)|6pr p−1 + k
;
pr p−1 − k
kr
kr
1+B
1+B
where n is de ned 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 de ned 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
!
n z +
n=k
=z p −
∞
X
∞
X
1+B
n=k
∞
X
1+B
B−A
n=k
−
p
−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,
f(z) =
∞
X
n fn (z) =
k−1 fk−1 +
n=k−1
n fn (z)
n=k
= 1−
!
∞
X
n z p +
n=k
= zp −
∞
X
∞
X
B−A
n=k
1+B
∞
X
n=k
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
n
!1=(p−n)
(4.1)
are de ned 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
′′
1 + zf (z) − p ¡ p; z ∈ U(r):
(4.2)
′
f (z)
Since
i P∞ 2
′′
n−1
e
zf
(z)
(n
−
np)|a
|z
n
n=k
1 +
P
− p = p−1
n−1
pz
f′ (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 de ned 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 de ned 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);
n
be de ned by (2.2) and (2.3). If
2
k
p2
(4.8)
k (; A; B)61;
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
k
!1=(p−k)
:
(4.10)
Proof. Let f ∈ T ( ; ). By Theorem 4.1 the function f is convex in U(rc ), where rc is de ned
by (4.1). For 6 − 2 the sequence {bn }, de ned 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
1=(p−n)
n
:
n
p
are de ned by (2.2) and (2.3).
r ∗ = inf (; A; B)
n¿k
and (; A; B);
n
(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
′
zf (z)
¡ p; z ∈ U(r):
−
p
(4.12)
f(z)
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J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255
Since
′
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 de ned 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 de ned 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
∗
k
1=(k−p)
Corollary 4.4. If 6 − 1; then
T ( ; ) ⊂ Sp∗ :
:
J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255
255
References
[1] J. Dziok, Classes of p-valent analytic functions with xed argument of coecients, I, to appear.
[2] J. Dziok, Classes of p-valent analytic functions with xed 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.
Classes of functions de ned by certain di erential–integral
operators
J. Dziok ∗
Institute of Mathematics, Pedagogical University of Rzeszow, ul. Rejtana 16A, PL-35-310 Rzeszow, 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 xed argument of coecient, which is de ned
in terms of certain di erential–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 +
zf′′ (z)
¿ 0; z ∈ U(r)
f′ (z)
(1.2)
and we say that a function f ∈ A(p; p + 1) is starlike in U(r); r ∈ (0; 1]; if
zf′ (z)
¿ 0;
Re
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)
246
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:
De nition 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 de ned for ¿ 0 and the function f by
Q f(z) =
1
( )
Z z
0
1−
z
−1
−1 f() d:
Now, we de ne the di erential–integral operator f(z) for 60: Then there exists a nonnegative
integer m and a real number
; −1 ¡
60; such that =
− m:
De nition 1.2. Let f ∈ A(p; k) and let log(z − ) be real when z − ¿ 0; z; ∈ U: The integral
operator f(z) is de ned 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
z
(z − )
−1 f() d:
0
The multiplicities of (z − ) −1 ; (z − )
in De nitions 1.1 and 1.2 are removed by requiring
log(z − ) ∈ R when z − ¿ 0:
By using these de nitions we de ne the linear operator
: A(p; k) → A(p; k)
by
f(z) =
(p + + ) −
z Q f(z) for ¿ 0
(p + )
f(z) =
(p + + ) 1− −
z
f(z) for 60:
(p + )
and
Putting = 1 in De nitions 1.1 and 1.2 we obtain the integral operator de ned by Owa [5] and
investigated by Srivastava and Owa [7], Dziok [1, 2] and others.
De nition 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
De nition 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 De nition 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 +
∞
(p + + ) X
(n + )
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
−1
n |an |6(; A; B);
(2.1)
n=k
where
and
B−A
(; A; B) = p
2
1 − B sin2 − B cos
n
=
(n + + ) (p + )
:
(n + ) (p + + )
Proof. Let f ∈ T ( ; ). By De nition 1.4 we obtain
f(z) 1 + A!(z)
=
;
zp
1 + B!(z)
(2.2)
(2.3)
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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
n−p
−1
¡ B
n |an |z
− A + Be
i
∞
X
n=k
n−p
−1
;
n |an |z
is de ned by (2.3). Thus, putting z = r; 0 ¡ r ¡ 1; we have
|w| ¡ |B − A + Bwei |;
(2.4)
where
w=
∞
X
n−p
−1
:
n |an |r
n=k
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−p
−1
n |an |r
¡ (; A; B);
n=k
where (; A; B);
n
are de ned 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 de ned 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)
J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255
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 = k; k + 1; k + 2; : : : ;
n;
are de ned 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 = k; k + 1; k + 2; : : : ;
n;
is de ned 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|f′ (z)|6pr p−1 + k(; A; B)
k−1
(3.2)
;
are de ned 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)
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J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255
Since
∞
∞
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
p
n
|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
6|f′ (z)|6pr p−1 + k
;
pr p−1 − k
kr
kr
1+B
1+B
where n is de ned 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 de ned 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
!
n z +
n=k
=z p −
∞
X
∞
X
1+B
n=k
∞
X
1+B
B−A
n=k
−
p
−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,
f(z) =
∞
X
n fn (z) =
k−1 fk−1 +
n=k−1
n fn (z)
n=k
= 1−
!
∞
X
n z p +
n=k
= zp −
∞
X
∞
X
B−A
n=k
1+B
∞
X
n=k
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
n
!1=(p−n)
(4.1)
are de ned 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
′′
1 + zf (z) − p ¡ p; z ∈ U(r):
(4.2)
′
f (z)
Since
i P∞ 2
′′
n−1
e
zf
(z)
(n
−
np)|a
|z
n
n=k
1 +
P
− p = p−1
n−1
pz
f′ (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 de ned 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 de ned 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);
n
be de ned by (2.2) and (2.3). If
2
k
p2
(4.8)
k (; A; B)61;
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
k
!1=(p−k)
:
(4.10)
Proof. Let f ∈ T ( ; ). By Theorem 4.1 the function f is convex in U(rc ), where rc is de ned
by (4.1). For 6 − 2 the sequence {bn }, de ned 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
1=(p−n)
n
:
n
p
are de ned by (2.2) and (2.3).
r ∗ = inf (; A; B)
n¿k
and (; A; B);
n
(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
′
zf (z)
¡ p; z ∈ U(r):
−
p
(4.12)
f(z)
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J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255
Since
′
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 de ned 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 de ned 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
∗
k
1=(k−p)
Corollary 4.4. If 6 − 1; then
T ( ; ) ⊂ Sp∗ :
:
J. Dziok / Journal of Computational and Applied Mathematics 105 (1999) 245–255
255
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