Paper32-Acta26-2011. 124KB Jun 04 2011 12:03:17 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 26/2011
pp. 321-330
INTEGRAL MEANS OF CERTAIN CLASSES OF ANALYTIC
FUNCTIONS DEFINED BY DZIOK-SRIVASTAVA OPERATOR
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
Abstract. In this paper, we introduce the subclass U Tq,s ([α1 ] ; α, β) of analytic
functions defined by Dziok-Srivastava operator. The object of the present paper is
to determine the Silvermen, s conjecture for the integral means inequality to this
class.
2000 Mathematics Subject Classification: 30C45..
1.
Introduction
Let A denote the class of functions of the form
f (z) = z +
∞
X
an z n , (1.1)
n=2
which are analytic in the open unit disc U = {z : |z| < 1}. Let K(α) and S ∗ (α)
denote the subclasses of A which are, respectively, convex and starlike functions of
order α, 0 ≤ α < 1. For convenience, we write K(0) = K and S ∗ (0) = S ∗ (see [18]).
The Hadamard product (or convolution) (f ∗g)(z) of the functions f (z) and g(z),
n
that is, if f (z) is given by (1.1) and g(z) is given by g(z) = z +∞
n=2 bn z , is defined
by
n
(f ∗ g)(z) = z +∞
n=2 an bn z = (g ∗ f )(z).
If f and g are analytic functions in U , we say that f is subordinate to g, written
f ≺ g if there exists a Schwarz function w, which (by definition) is analytic in U
with w(0) = 0 and |w(z)| < 1 for all z ∈ U, such that f (z) = g(w(z)), z ∈ U.
−
For positive real parameters α1 , ..., αq and β1 , ..., βs (βj ∈ C\Z−
0 , Z0 = 0, −1, −2,
...; j = 1, 2, ..., s) , the generalized hypergeometric function q Fs (α1 , ...., αq ; β1 , ..., βs ; z) is
defined by
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
∞
X
(α1 )n ...(αq )n n
z
q Fs (α1 , ....., αq ; β1 , ....., βs ; z) =
(β1 )n ...(βs )n n!
n=0
(q ≤ s + 1; s, q ∈ N0 = N ∪ {0}, N = {1, 2, .........}; z ∈ U ),
where (θ)n , is the Pochhammer symbol defined in terms of the Gamma function
Γ, by
Γ(θ + n)
1
(n = 0)
(θ)n =
=
θ(θ + 1)....(θ + n − 1)
(n ∈ N).
Γ(θ)
For the function h(α1 , ...., αq ; β1 , ...βs ; z) = zq Fs (α1 , ....., αq ; β1 , ....., βs ; z),
the Dziok-Srivastava linear operator ( see [5] and [6] ) Hq,s (α1 , ....., αq ; β1 , ...
..., βs ) : A −→ A, is defined by the Hadamard product as follows:
Hq,s (α1 , ....., αq ; β1 , ....., βs )f (z) = h(α1 , ...., αq ; β1 , ...βs ; z) ∗ f (z)
∞
X
= z+
Ψn (α1 ) an z n (z ∈ U ),
(1.2)
n=2
where
Ψn (α1 ) = (α1 )n−1 .......(αq )n−1 (β1 )n−1 ...(βs )n−1 (n − 1)!. (1.3)
For brevity, we write
Hq,s (α1 , ....., αq ; β1 , ....., βs ; z)f (z) = Hq,s (α1 )f (z). (1.4)
For 0 ≤ α < 1, β ≥ 0 and for all z ∈ U, let U Sq,s ([α1 ] ; α, β) denote the subclass of
A consisting
of functions fo(z) of the form (1.1) and satisfying the analytic criterion
n
H (α )f (z)
Hq,s (α1 )f (z)
Re z(H (α )f (z))′ − α > β z(Hq,s (α1 )f (z))′ − 1 . (1.5)
q,s
q,s
1
1
Denote by T the subclass of A consisting of functions of the form:
∞
P
f(z)=zan z n (an ≥ 0),
(1.6)
n=2
which are analytic in U . We define the class U Tq,s ([α1 ] ; α, β) by:
UTq,s ([α1 ] ; α, β) = U Sq,s ([α1 ] ; α, β) ∩ T. (1.7)
We note that for suitable choices of q, s, α and β, we obtain the following subclasses
studied by various authors.
(1) For q = 2 and s = α1 = α2 = β1 = 1 in (1.5), the class U T2,1 ([1] ; α, β)
reduces to the class ST (α, β)
f (z)
f (z)
= f ∈ T : Re
− α > β ′
− 1 , 0 ≤ α < 1, β ≥ 0, z ∈ U }
′
zf (z)
zf (z)
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
and the class ST (α, 0) = ST (α) is the family of functions f (z) ∈ T which satisfy
the following condition (see [7] and [19])
f (z)
ST (α) = Re
> α (0 ≤ α < 1);
zf ′ (z)
(2) For q = 2, s = 1, α1 = a (a > 0) , α2 = 1 and β1 = c (c > 0) in (1.5), the
class U T2,1 ([a, 1; c] ; α, β) reduces to the class T (a, c; α, β)
L(a, c)f (z)
L(a, c)f (z)
>β
=
f ∈ T : Re
′ − α
′ − 1 , 0 ≤ α <
z(L(a, c)f (z))
z(L(a, c)f (z))
1, β ≥ 0, z ∈ U } ,
where L(a, c) is the Carlson - Shaffer operator (see [2]);
(3) For q = 2, s = 1, α1 = λ + 1(λ > −1) and α2 = β1 = 1 in (1.5), the
class U T2,1 ([λ + 1]; α, β) reduces to the class Wλ (α, β)
=
Dλ f (z)
Dλ f (z)
>β
f ∈ T : Re
′ − α
′ − 1 , 0 ≤ α <
λ
λ
z(D f (z))
z(D f (z))
1, β ≥ 0, λ > −1, z ∈ U } ( see [11] ),
where Dλ (λ > −1) is the Ruscheweyh derivative operator (see [15]);
(4) For q = 2, s = 1, α1 = v + 1 (v > −1) , α2 = 1 and β1 = v + 2 in (1.5), the
class U T2,1 ([v + 1, 1; v + 2] ; α, β) reduces to the class ζT (v; α, β)
Jv f (z)
Jv f (z)
=
f ∈ T : Re
> β
′ − α
′ − 1 , 0 ≤ α < 1, β ≥
z(Jv f (z))
z(Jv f (z))
0, v > −1, z ∈ U } ,
where Jv f (z) is the generalized Bernardi - Libera - Livingston operator (see [1], [8]
and [10]);
(5) For q = 2, s = 1, α1 = 2, α2 = 1 and β1 = 2 − µ (µ 6= 2, 3, ....) in (1.5), the
class U T2,1 ([2, 1; 2 − µ]; α, β) reduces to the class T (µ; α, β)
Ωµz f (z)
Ωµz f (z)
>β
=
f ∈ T : Re
′ − α
′ − 1 , 0 ≤ α < 1, β
µ
µ
z(Ωz f (z))
z(Ωz f (z))
≥ 0, µ 6= 2, 3, ...., z ∈ U } ,
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
where Ωµz f (z) is the Srivastava - Owa fractional derivative operator (see [13] and
[14]);
(6) For q = 2, s = 1, α1 = µ (µ > 0) , α2 = 1 and β1 = λ + 1 (λ > −1) in (1.5), the
class U T2,1 ([µ, 1; λ + 1]; α, β) reduces to the class £T (µ, λ; α, β)
Iλ,µ f (z)
Iλ,µ f (z)
=
f ∈ T : Re
> β
′ − α
′ − 1 , −1 ≤ α < 1,
z(Iλ,µ f (z))
z(Iλ,µ f (z))
β ≥ 0, µ > 0, λ > −1, z ∈ U } ,
where Iλ,µ f (z) is the Choi-Saigo-Srivastava operator (see [4]);
(7) For q = 2, s = 1, α1 = 2, α2 = 1 and β1 = k + 1 (k > −1) in (1.5), the
class U T2,1 ([2, 1; k + 1]; α, β) reduces to the class AT (k; α, β)
=
Ik f (z)
−α
f ∈ T : Re
z(Ik f (z))′
β ≥ 0, k > −1, z ∈ U } ,
Ik f (z)
>β
′ − 1 , 0 ≤ α < 1,
z(I f (z))
k
where Ik f (z) is the Noor integral operator (see [12]);
(8) For q = 2, s = 1, α1 = c (c > 0) , α2 = λ + 1 (λ > −1) and β1 = a (a > 0) in (1.5),
the class U T2,1 ([c, λ + 1; a] ; α, β) reduces to the class ̥T (c, a, λ; α, β)
λ
λ
I (a, c)f (z)
I (a, c)f (z)
, 0 ≤
−
α
>
β
−
1
=
f ∈ T : Re
′
′
z(I λ (a, c)f (z))
z(I λ (a, c)f (z))
α < 1, β ≥ 0, c > 0, λ > −1, a > 0, z ∈ U } ,
where I λ (a, c)f (z) is the Cho-Kwon-Srivastava operator (see [3]).
In [16] Silverman found that the function f2 = z − z 2 2 is often extremal over
the family T . He applied this function to resolve his integral means inequality,
conjectured and settled in [17]:
Z 2π
δ
δ
iθ
iθ
2π
f
(re
)
dθ
≤
)
f
(re
dθ,
2
0
0
for all f ∈ T , δ > 0 and 0 < r < 1. In [17], he also proved his conjecture for the
subclasses T ∗ (α) and C(α) of T , where C(α) and T ∗ (α) are the classes of convex
and starlike functions of order α, 0 ≤ α < 1, respectively.
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
In this paper, we prove Silverman, s conjecture for functions in the class U Sq,s ([α1 ] ;
α, β). Also by taking appropriate choices of the parameters α1 , ..., αq and β1 , ..., βs ,
we obtain the integral means inequalities for several known as well as new subclasses
of uniformly convex and uniformly starlike functions in U .
2. Coefficient estimates
Unless otherwise mentioned, we shall assume in the reminder of this paper that,
the parameters α1 , ..., αq and β1 , ..., βs are positive real numbers,−1 ≤ α < 1, β ≥
0, n ≥ 2, z ∈ U and Ψn (α1 ) is defined by (1.3).
Theorem 1. A function f (z) of the form (1.6) is in the class U Tq,s ([α1 ] ; α, β)
if
∞ [2n − n(α − β) − (β + 1)]Ψ (α ) a ≤ 1 − α.
(2.1)
n
1
n
n=2
Proof. Suppose that (2.1) is true. Since
[2n − n(α − β) − (β + 1)]Ψn (α1 )
(n − 1) (1 + β) Ψn (α1 )
− nΨn (α1 ) =
> 0,
1−α
1−α
we deduce
∞
n=2 nΨn (α1 ) an
(1 − α) −∞
n=2 [2n − n(α − β) − (β + 1)]Ψn (α1 ) an 1 −n=2 nΨn (α1 ) an ≥ 0.
325
(2.2)
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
This completes the proof of Theorem 1.
Unfortunately, the converse of the above Theorem 1 is not true. So we define
the subclass Tq,s ([α1 ] ; α, β) of U Tq,s ([α1 ] ; α, β) consisting of functions f (z) which
satisfy (2.1).
Remark 1. Putting q = 2, s = 1, β = 0 and α1 = α2 = β1 = 1, in Theorem 1, we
will obtain the result obtained by Yamakawa [19, Lemma 2.1, with n = p = 1].
Corollary 1. Let the function f (z) defined by (1.6) be in the class Tq,s ([α1 ] ; α, β),
then
an ≤ (1 − α)[2n − n(α − β) − (β + 1)]Ψn (α1 ) (n ≥ 2). (2.3)The result is sharp
for the function f(z)=z-(1 − α) [2n − n(α − β) − (β + 1)]Ψn (α1 )z n (n ≥ 2). (2.4)
Putting q = 2, s = 1, α1 = λ + 1(λ > −1) and α2 = β1 = 1 in Theorem 1, we obtain
the following corollary.
Corollary 2. A function f (z) of the form (1.6) is in the class Wλ (α, β)
if
(λ + 1)n−1
∞
an ≤ 1 − α.
n=2 [2n − n(α − β) − (β + 1)]
(n − 1)!
Remark 2. The result in Corollary 2 correct the result obtained by Najafzadeh
and Kulkarni [11, Lemma 1.1].
3.Integral Means
Lemma 1 [9]. If the functions f and g are analytic in U with g ≺ f , then
for δ > 0 and 0 < r < 1,
Z 2π
δ
δ
iθ
iθ
2π
f
(re
)
dθ
≤
g(re
)
dθ.
0
0
Applying Theorem 1 and Lemma 1 we prove the following theorem.
Theorem 2. Suppose f (z) ∈ Tq,s ([α1 ] ; α, β), δ > 0, the sequence {Ψn (α1 )} (n ≥ 2) is
non-decrecing and f2 (z) is defined by:
f2 (z) = z − 1 − α(3 − 2α + β)Ψ2 (α1 )z 2 , (3.1)
then for z = reiθ , 0 < r < 1, we have
R
2π f (reiθ ) δ dθ ≤ 2π f (reiθ ) δ dθ.
(3.2)
2
0
0
Proof. For f (z) of the form (1.6), (3.2) is equivalent to proving that
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
2π
0
n−1 δ
1 −∞
dθ ≤
n=2 an z
Z
2π
|1 − (1 − α)(3 − 2α + β)Ψ2 (α1 )z|δ dθ.
0
By using Lemma 1, it suffices to show that
n−1 ≺ 1 − (1 − α)(3 − 2α + β)Ψ (α )z.
1-∞
(3.3)
2
1
n=2 an z
Setting
n−1 = 1 − (1 − α)(3 − 2α + β)Ψ (α )w(z),
1-∞
(3.4)
2
1
n=2 an z
and using (2.1) and the hypothises {Ψn (α1 )} (n ≥ 2) is non-decrecing, we obtain
n−1
|w(z)| = (3 − 2α + β)Ψ2 (α1 )(1 − α)∞
n=2 an z
≤ |z|∞
n=2 (3 − 2α + β)Ψ2 (α1 )(1 − α)an
[2n − n(α − β) − (β + 1)]Ψn (α1 )
≤ |z|∞
an
n=2
(1 − α)
≤ |z| .
This completes the proof of Theorem 2.
Putting q = 2 and s = α1 = α2 = β1 = 1 in Theorems 1 and 2, respectively, we
obtain the following corollary:
Corollary 3. If f (z) ∈ ST (α, β), δ > 0, then the assertion (3.2) holds true, where
f2 (z) = z − (1 − α)(3 − 2α + β)z 2 .
Putting β = 0 in Corollary 3, we obtain the following corollary:
Corollary 4. If f (z) ∈ ST (α), δ > 0, then the assertion (3.2) holds true, where
f2 (z) = z − (1 − α)(3 − 2α)z 2 .
Putting q = 2, s = 1, α1 = a (a > 0), α2 = 1 and β1 = c (c > 0) in Theorems 1
and 2, respectively, we obtain the following corollary:
Corollary 5. If f (z) ∈ T (a, c; α, β), δ > 0, then the assertion (3.2) holds true,
where
f2 (z) = z − (1 − α)c(3 − 2α + β)az 2 .
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
Putting q = 2, s = 1, α1 = λ + 1 (λ > −1) and α2 = β1 = 1 in Theorems 1 and 2,
respectively, we obtain the following corollary:
Corollary 6. If f (z) ∈ Wλ (α, β), δ > 0, then the assertion (3.2) holds true, where
f2 (z) = z − (1 − α)(3 − 2α + β)(λ + 1)z 2 .
Putting q = 2, s = 1, α1 = v + 1 (v > −1), α2 = 1 and β1 = v + 2 in Theorems 1
and 2, respectively, we obtain the following corollary:
Corollary 7. If f (z) ∈ ζT (v; α, β), δ > 0, then the assertion (3.2) holds true,
where
f2 (z) = z − (1 − α)(v + 2)(3 − 2α + β)(v + 1)z 2 .
Putting q = 2, s = 1, α1 = 2, α2 = 1 and β1 = 2 − µ (µ 6= 2, 3, ....) in Theorems 1
and 2, respectively, we obtain the following corollary:
Corollary 8. If f (z) ∈ T (µ; α, β), δ > 0, then the assertion (3.2) holds true, where
f2 (z) = z − (1 − α)(2 − µ)2(3 − 2α + β)z 2 .
Putting q = 2, s = 1, α1 = µ(µ > 0), α2 = 1 and β1 = λ + 1(λ > −1) in Theorems
1 and 2, respectively, we obtain the following corollary:
Corollary 9. If f (z) ∈ £T (µ, λ; α, β), δ > 0, then the assertion (3.2) holds true,
where
f2 (z) = z − (1 − α)(λ + 1)µ(3 − 2α + β)z 2 .
Putting q = 2, s = 1, α1 = 2, α2 = 1 and β1 = k + 1(k > −1) in Theorems 1 and
2, respectively, we obtain the following corollary:
Corollary 10. If f (z) ∈ AT (k; α, β), δ > 0, then the assertion (3.2) holds true,
where
f2 (z) = z − (1 − α)(k + 1)2(3 − 2α + β)z 2 .
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
Putting q = 3, s = 2, α1 = c, α2 = λ + 1 and β1 = a in Theorems 1 and 2,
respectively, we obtain the following corollary:
Corollary 11. If f (z) ∈ ̥T (c, λ; a; α, β), δ > 0, then the assertion (3.2) holds
true, where
f2 (z) = z − a(1 − α)c(λ + 1)(3 − 2α + β)z 2 .
References
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Soc., 135 (1969), 429–449.
[2] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric
functions, J. Math. Anal. Appl., 15 (1984), 737-745.
[3] N. E. Cho, O. S. Kwon and H. M. Srivastava, Inclusion relationships and
argument properties for certain subclasses of multivalent functions associated with
a family of linear operators, J. Math. Anal. Appl., 292(2004), 470-483.
[4] J. H. Choi, M. Saigo and H. M. Srivastava, Some inclusion properties of a
certain family of integral operators, J. Math. Anal. Appl., 276(2002), 432-445.
[5] J. Dziok and H.M. Srivastava, Classes of analytic functions with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), 1-13.
[6] J. Dziok and H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec.
Funct., 14 (2003), 7-18.
[7] J. S. Kang, S. Owa and H. M. Srivastava, Quasi-convolution properties of
certain subclasses of analytic functions, Bull. Belg. Math. Soc., 3 (1996), 603-608.
[8] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math.
Soc., 16 (1969), 755-758.
[9] J. L. Littlewood, On inequalities in the theory functions, Proc. London Math.
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[10] A. E. Livingston, On the radius of univalence of certain analytic functions,
Proc. Amer. Math. Soc., 17 (1966), 352-357.
[11] S. Najafzadeh and S. R. Kulkarni, Study of certain properties of univalent
functions based on Ruscheweyh derivative, South. Asian Bull. Math., 32 (2008),
475-488.
[12] K. I. Noor, On new classes of integral operators, J. Natur. Geom., 16(1999),
71-80.
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[13] S. Owa, On the distortion theorems. I, Kyungpook Math. J., 18 (1978),
53-59.
[14] S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39 (1987), 1057-1077.
[15] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math.
Soc., 49 (1975), 109-115.
[16] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., 51 (1975), 109-116.
[17] H. Silverman, Integral means for univalent functions with negative coefficients, Houston J. Math., 23 (1997), 169-174.
[18] H.M. Srivastava and S. Owa (Editors), Current Topics in Analytic Function
Theory, World Scientific Publishing Company, Singapore, New Jersey, London, Hong
Kong, 1992.
[19] R. Yamakawa, Certain subclasses of p-valently starlike functions with negative coefficients, in: Current Topics in Analytic Function Theory, World Scientific
Publishing Company, Singapore, New Jersey, London, Hong Kong, 1992
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
Department of Mathematics
Faculty of Science
Mansoura University
Mansoura 35516, Egypt.
email: [email protected], [email protected], [email protected],
[email protected]
330
ISSN: 1582-5329
No. 26/2011
pp. 321-330
INTEGRAL MEANS OF CERTAIN CLASSES OF ANALYTIC
FUNCTIONS DEFINED BY DZIOK-SRIVASTAVA OPERATOR
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
Abstract. In this paper, we introduce the subclass U Tq,s ([α1 ] ; α, β) of analytic
functions defined by Dziok-Srivastava operator. The object of the present paper is
to determine the Silvermen, s conjecture for the integral means inequality to this
class.
2000 Mathematics Subject Classification: 30C45..
1.
Introduction
Let A denote the class of functions of the form
f (z) = z +
∞
X
an z n , (1.1)
n=2
which are analytic in the open unit disc U = {z : |z| < 1}. Let K(α) and S ∗ (α)
denote the subclasses of A which are, respectively, convex and starlike functions of
order α, 0 ≤ α < 1. For convenience, we write K(0) = K and S ∗ (0) = S ∗ (see [18]).
The Hadamard product (or convolution) (f ∗g)(z) of the functions f (z) and g(z),
n
that is, if f (z) is given by (1.1) and g(z) is given by g(z) = z +∞
n=2 bn z , is defined
by
n
(f ∗ g)(z) = z +∞
n=2 an bn z = (g ∗ f )(z).
If f and g are analytic functions in U , we say that f is subordinate to g, written
f ≺ g if there exists a Schwarz function w, which (by definition) is analytic in U
with w(0) = 0 and |w(z)| < 1 for all z ∈ U, such that f (z) = g(w(z)), z ∈ U.
−
For positive real parameters α1 , ..., αq and β1 , ..., βs (βj ∈ C\Z−
0 , Z0 = 0, −1, −2,
...; j = 1, 2, ..., s) , the generalized hypergeometric function q Fs (α1 , ...., αq ; β1 , ..., βs ; z) is
defined by
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
∞
X
(α1 )n ...(αq )n n
z
q Fs (α1 , ....., αq ; β1 , ....., βs ; z) =
(β1 )n ...(βs )n n!
n=0
(q ≤ s + 1; s, q ∈ N0 = N ∪ {0}, N = {1, 2, .........}; z ∈ U ),
where (θ)n , is the Pochhammer symbol defined in terms of the Gamma function
Γ, by
Γ(θ + n)
1
(n = 0)
(θ)n =
=
θ(θ + 1)....(θ + n − 1)
(n ∈ N).
Γ(θ)
For the function h(α1 , ...., αq ; β1 , ...βs ; z) = zq Fs (α1 , ....., αq ; β1 , ....., βs ; z),
the Dziok-Srivastava linear operator ( see [5] and [6] ) Hq,s (α1 , ....., αq ; β1 , ...
..., βs ) : A −→ A, is defined by the Hadamard product as follows:
Hq,s (α1 , ....., αq ; β1 , ....., βs )f (z) = h(α1 , ...., αq ; β1 , ...βs ; z) ∗ f (z)
∞
X
= z+
Ψn (α1 ) an z n (z ∈ U ),
(1.2)
n=2
where
Ψn (α1 ) = (α1 )n−1 .......(αq )n−1 (β1 )n−1 ...(βs )n−1 (n − 1)!. (1.3)
For brevity, we write
Hq,s (α1 , ....., αq ; β1 , ....., βs ; z)f (z) = Hq,s (α1 )f (z). (1.4)
For 0 ≤ α < 1, β ≥ 0 and for all z ∈ U, let U Sq,s ([α1 ] ; α, β) denote the subclass of
A consisting
of functions fo(z) of the form (1.1) and satisfying the analytic criterion
n
H (α )f (z)
Hq,s (α1 )f (z)
Re z(H (α )f (z))′ − α > β z(Hq,s (α1 )f (z))′ − 1 . (1.5)
q,s
q,s
1
1
Denote by T the subclass of A consisting of functions of the form:
∞
P
f(z)=zan z n (an ≥ 0),
(1.6)
n=2
which are analytic in U . We define the class U Tq,s ([α1 ] ; α, β) by:
UTq,s ([α1 ] ; α, β) = U Sq,s ([α1 ] ; α, β) ∩ T. (1.7)
We note that for suitable choices of q, s, α and β, we obtain the following subclasses
studied by various authors.
(1) For q = 2 and s = α1 = α2 = β1 = 1 in (1.5), the class U T2,1 ([1] ; α, β)
reduces to the class ST (α, β)
f (z)
f (z)
= f ∈ T : Re
− α > β ′
− 1 , 0 ≤ α < 1, β ≥ 0, z ∈ U }
′
zf (z)
zf (z)
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
and the class ST (α, 0) = ST (α) is the family of functions f (z) ∈ T which satisfy
the following condition (see [7] and [19])
f (z)
ST (α) = Re
> α (0 ≤ α < 1);
zf ′ (z)
(2) For q = 2, s = 1, α1 = a (a > 0) , α2 = 1 and β1 = c (c > 0) in (1.5), the
class U T2,1 ([a, 1; c] ; α, β) reduces to the class T (a, c; α, β)
L(a, c)f (z)
L(a, c)f (z)
>β
=
f ∈ T : Re
′ − α
′ − 1 , 0 ≤ α <
z(L(a, c)f (z))
z(L(a, c)f (z))
1, β ≥ 0, z ∈ U } ,
where L(a, c) is the Carlson - Shaffer operator (see [2]);
(3) For q = 2, s = 1, α1 = λ + 1(λ > −1) and α2 = β1 = 1 in (1.5), the
class U T2,1 ([λ + 1]; α, β) reduces to the class Wλ (α, β)
=
Dλ f (z)
Dλ f (z)
>β
f ∈ T : Re
′ − α
′ − 1 , 0 ≤ α <
λ
λ
z(D f (z))
z(D f (z))
1, β ≥ 0, λ > −1, z ∈ U } ( see [11] ),
where Dλ (λ > −1) is the Ruscheweyh derivative operator (see [15]);
(4) For q = 2, s = 1, α1 = v + 1 (v > −1) , α2 = 1 and β1 = v + 2 in (1.5), the
class U T2,1 ([v + 1, 1; v + 2] ; α, β) reduces to the class ζT (v; α, β)
Jv f (z)
Jv f (z)
=
f ∈ T : Re
> β
′ − α
′ − 1 , 0 ≤ α < 1, β ≥
z(Jv f (z))
z(Jv f (z))
0, v > −1, z ∈ U } ,
where Jv f (z) is the generalized Bernardi - Libera - Livingston operator (see [1], [8]
and [10]);
(5) For q = 2, s = 1, α1 = 2, α2 = 1 and β1 = 2 − µ (µ 6= 2, 3, ....) in (1.5), the
class U T2,1 ([2, 1; 2 − µ]; α, β) reduces to the class T (µ; α, β)
Ωµz f (z)
Ωµz f (z)
>β
=
f ∈ T : Re
′ − α
′ − 1 , 0 ≤ α < 1, β
µ
µ
z(Ωz f (z))
z(Ωz f (z))
≥ 0, µ 6= 2, 3, ...., z ∈ U } ,
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
where Ωµz f (z) is the Srivastava - Owa fractional derivative operator (see [13] and
[14]);
(6) For q = 2, s = 1, α1 = µ (µ > 0) , α2 = 1 and β1 = λ + 1 (λ > −1) in (1.5), the
class U T2,1 ([µ, 1; λ + 1]; α, β) reduces to the class £T (µ, λ; α, β)
Iλ,µ f (z)
Iλ,µ f (z)
=
f ∈ T : Re
> β
′ − α
′ − 1 , −1 ≤ α < 1,
z(Iλ,µ f (z))
z(Iλ,µ f (z))
β ≥ 0, µ > 0, λ > −1, z ∈ U } ,
where Iλ,µ f (z) is the Choi-Saigo-Srivastava operator (see [4]);
(7) For q = 2, s = 1, α1 = 2, α2 = 1 and β1 = k + 1 (k > −1) in (1.5), the
class U T2,1 ([2, 1; k + 1]; α, β) reduces to the class AT (k; α, β)
=
Ik f (z)
−α
f ∈ T : Re
z(Ik f (z))′
β ≥ 0, k > −1, z ∈ U } ,
Ik f (z)
>β
′ − 1 , 0 ≤ α < 1,
z(I f (z))
k
where Ik f (z) is the Noor integral operator (see [12]);
(8) For q = 2, s = 1, α1 = c (c > 0) , α2 = λ + 1 (λ > −1) and β1 = a (a > 0) in (1.5),
the class U T2,1 ([c, λ + 1; a] ; α, β) reduces to the class ̥T (c, a, λ; α, β)
λ
λ
I (a, c)f (z)
I (a, c)f (z)
, 0 ≤
−
α
>
β
−
1
=
f ∈ T : Re
′
′
z(I λ (a, c)f (z))
z(I λ (a, c)f (z))
α < 1, β ≥ 0, c > 0, λ > −1, a > 0, z ∈ U } ,
where I λ (a, c)f (z) is the Cho-Kwon-Srivastava operator (see [3]).
In [16] Silverman found that the function f2 = z − z 2 2 is often extremal over
the family T . He applied this function to resolve his integral means inequality,
conjectured and settled in [17]:
Z 2π
δ
δ
iθ
iθ
2π
f
(re
)
dθ
≤
)
f
(re
dθ,
2
0
0
for all f ∈ T , δ > 0 and 0 < r < 1. In [17], he also proved his conjecture for the
subclasses T ∗ (α) and C(α) of T , where C(α) and T ∗ (α) are the classes of convex
and starlike functions of order α, 0 ≤ α < 1, respectively.
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
In this paper, we prove Silverman, s conjecture for functions in the class U Sq,s ([α1 ] ;
α, β). Also by taking appropriate choices of the parameters α1 , ..., αq and β1 , ..., βs ,
we obtain the integral means inequalities for several known as well as new subclasses
of uniformly convex and uniformly starlike functions in U .
2. Coefficient estimates
Unless otherwise mentioned, we shall assume in the reminder of this paper that,
the parameters α1 , ..., αq and β1 , ..., βs are positive real numbers,−1 ≤ α < 1, β ≥
0, n ≥ 2, z ∈ U and Ψn (α1 ) is defined by (1.3).
Theorem 1. A function f (z) of the form (1.6) is in the class U Tq,s ([α1 ] ; α, β)
if
∞ [2n − n(α − β) − (β + 1)]Ψ (α ) a ≤ 1 − α.
(2.1)
n
1
n
n=2
Proof. Suppose that (2.1) is true. Since
[2n − n(α − β) − (β + 1)]Ψn (α1 )
(n − 1) (1 + β) Ψn (α1 )
− nΨn (α1 ) =
> 0,
1−α
1−α
we deduce
∞
n=2 nΨn (α1 ) an
(1 − α) −∞
n=2 [2n − n(α − β) − (β + 1)]Ψn (α1 ) an 1 −n=2 nΨn (α1 ) an ≥ 0.
325
(2.2)
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
This completes the proof of Theorem 1.
Unfortunately, the converse of the above Theorem 1 is not true. So we define
the subclass Tq,s ([α1 ] ; α, β) of U Tq,s ([α1 ] ; α, β) consisting of functions f (z) which
satisfy (2.1).
Remark 1. Putting q = 2, s = 1, β = 0 and α1 = α2 = β1 = 1, in Theorem 1, we
will obtain the result obtained by Yamakawa [19, Lemma 2.1, with n = p = 1].
Corollary 1. Let the function f (z) defined by (1.6) be in the class Tq,s ([α1 ] ; α, β),
then
an ≤ (1 − α)[2n − n(α − β) − (β + 1)]Ψn (α1 ) (n ≥ 2). (2.3)The result is sharp
for the function f(z)=z-(1 − α) [2n − n(α − β) − (β + 1)]Ψn (α1 )z n (n ≥ 2). (2.4)
Putting q = 2, s = 1, α1 = λ + 1(λ > −1) and α2 = β1 = 1 in Theorem 1, we obtain
the following corollary.
Corollary 2. A function f (z) of the form (1.6) is in the class Wλ (α, β)
if
(λ + 1)n−1
∞
an ≤ 1 − α.
n=2 [2n − n(α − β) − (β + 1)]
(n − 1)!
Remark 2. The result in Corollary 2 correct the result obtained by Najafzadeh
and Kulkarni [11, Lemma 1.1].
3.Integral Means
Lemma 1 [9]. If the functions f and g are analytic in U with g ≺ f , then
for δ > 0 and 0 < r < 1,
Z 2π
δ
δ
iθ
iθ
2π
f
(re
)
dθ
≤
g(re
)
dθ.
0
0
Applying Theorem 1 and Lemma 1 we prove the following theorem.
Theorem 2. Suppose f (z) ∈ Tq,s ([α1 ] ; α, β), δ > 0, the sequence {Ψn (α1 )} (n ≥ 2) is
non-decrecing and f2 (z) is defined by:
f2 (z) = z − 1 − α(3 − 2α + β)Ψ2 (α1 )z 2 , (3.1)
then for z = reiθ , 0 < r < 1, we have
R
2π f (reiθ ) δ dθ ≤ 2π f (reiθ ) δ dθ.
(3.2)
2
0
0
Proof. For f (z) of the form (1.6), (3.2) is equivalent to proving that
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
2π
0
n−1 δ
1 −∞
dθ ≤
n=2 an z
Z
2π
|1 − (1 − α)(3 − 2α + β)Ψ2 (α1 )z|δ dθ.
0
By using Lemma 1, it suffices to show that
n−1 ≺ 1 − (1 − α)(3 − 2α + β)Ψ (α )z.
1-∞
(3.3)
2
1
n=2 an z
Setting
n−1 = 1 − (1 − α)(3 − 2α + β)Ψ (α )w(z),
1-∞
(3.4)
2
1
n=2 an z
and using (2.1) and the hypothises {Ψn (α1 )} (n ≥ 2) is non-decrecing, we obtain
n−1
|w(z)| = (3 − 2α + β)Ψ2 (α1 )(1 − α)∞
n=2 an z
≤ |z|∞
n=2 (3 − 2α + β)Ψ2 (α1 )(1 − α)an
[2n − n(α − β) − (β + 1)]Ψn (α1 )
≤ |z|∞
an
n=2
(1 − α)
≤ |z| .
This completes the proof of Theorem 2.
Putting q = 2 and s = α1 = α2 = β1 = 1 in Theorems 1 and 2, respectively, we
obtain the following corollary:
Corollary 3. If f (z) ∈ ST (α, β), δ > 0, then the assertion (3.2) holds true, where
f2 (z) = z − (1 − α)(3 − 2α + β)z 2 .
Putting β = 0 in Corollary 3, we obtain the following corollary:
Corollary 4. If f (z) ∈ ST (α), δ > 0, then the assertion (3.2) holds true, where
f2 (z) = z − (1 − α)(3 − 2α)z 2 .
Putting q = 2, s = 1, α1 = a (a > 0), α2 = 1 and β1 = c (c > 0) in Theorems 1
and 2, respectively, we obtain the following corollary:
Corollary 5. If f (z) ∈ T (a, c; α, β), δ > 0, then the assertion (3.2) holds true,
where
f2 (z) = z − (1 − α)c(3 − 2α + β)az 2 .
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
Putting q = 2, s = 1, α1 = λ + 1 (λ > −1) and α2 = β1 = 1 in Theorems 1 and 2,
respectively, we obtain the following corollary:
Corollary 6. If f (z) ∈ Wλ (α, β), δ > 0, then the assertion (3.2) holds true, where
f2 (z) = z − (1 − α)(3 − 2α + β)(λ + 1)z 2 .
Putting q = 2, s = 1, α1 = v + 1 (v > −1), α2 = 1 and β1 = v + 2 in Theorems 1
and 2, respectively, we obtain the following corollary:
Corollary 7. If f (z) ∈ ζT (v; α, β), δ > 0, then the assertion (3.2) holds true,
where
f2 (z) = z − (1 − α)(v + 2)(3 − 2α + β)(v + 1)z 2 .
Putting q = 2, s = 1, α1 = 2, α2 = 1 and β1 = 2 − µ (µ 6= 2, 3, ....) in Theorems 1
and 2, respectively, we obtain the following corollary:
Corollary 8. If f (z) ∈ T (µ; α, β), δ > 0, then the assertion (3.2) holds true, where
f2 (z) = z − (1 − α)(2 − µ)2(3 − 2α + β)z 2 .
Putting q = 2, s = 1, α1 = µ(µ > 0), α2 = 1 and β1 = λ + 1(λ > −1) in Theorems
1 and 2, respectively, we obtain the following corollary:
Corollary 9. If f (z) ∈ £T (µ, λ; α, β), δ > 0, then the assertion (3.2) holds true,
where
f2 (z) = z − (1 − α)(λ + 1)µ(3 − 2α + β)z 2 .
Putting q = 2, s = 1, α1 = 2, α2 = 1 and β1 = k + 1(k > −1) in Theorems 1 and
2, respectively, we obtain the following corollary:
Corollary 10. If f (z) ∈ AT (k; α, β), δ > 0, then the assertion (3.2) holds true,
where
f2 (z) = z − (1 − α)(k + 1)2(3 − 2α + β)z 2 .
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan - Integral Means of...
Putting q = 3, s = 2, α1 = c, α2 = λ + 1 and β1 = a in Theorems 1 and 2,
respectively, we obtain the following corollary:
Corollary 11. If f (z) ∈ ̥T (c, λ; a; α, β), δ > 0, then the assertion (3.2) holds
true, where
f2 (z) = z − a(1 − α)c(λ + 1)(3 − 2α + β)z 2 .
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M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
Department of Mathematics
Faculty of Science
Mansoura University
Mansoura 35516, Egypt.
email: [email protected], [email protected], [email protected],
[email protected]
330