Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 153280, 10 pages
doi:10.1155/2008/153280
Research Article
Second Hankel Determinant for a Class of Analytic
Functions Defined by Fractional Derivative
A. K. Mishra and P. Gochhayat
Department of Mathematics, Berhampur University, Bhanja Bihar, Berhampur 760 007, Orissa, India
Correspondence should be addressed to A. K. Mishra, akshayam2001@yahoo.co.in
Received 4 April 2007; Accepted 21 November 2007
Recommended by Vladimir Mityushev
By making use of the fractional differential operator Ωλz due to Owa and Srivastava, a class of analytic functions Rλ α, ρ 0 ≤ ρ ≤ 1, 0 ≤ λ < 1, |α| < π/2 is introduced. The sharp bound for the
nonlinear functional |a2 a4 − a23 | is found. Several basic properties such as inclusion, subordination,
integral transform, Hadamard product are also studied.
Copyright q 2008 A. K. Mishra and P. Gochhayat. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let A denote the class of functions analytic in the open unit disc
U : z : z ∈ C, |z| < 1
1.1
and let A0 be the class of functions f in A given by the normalized power series
fz z ∞
an zn
z ∈ U.
1.2
n2
Also let S, S ∗ β, CVβ, and K denote, respectively, the subclasses of A0 consisting of
functions which are univalent, starlike of order β, convex of order β cf. 1, and close-to-convex
cf. 2 in U. In particular, S ∗ 0 S ∗ and CV0 CV are the familiar classes of starlike and
convex functions in U cf. 2.
Given f and g in A, the function f is said to be subordinate to g in U if there exits a function ω ∈ A satisfying the conditions of the Schwarz Lemma such that fz gωz, z ∈ U.
We denote the subordination by
fz ≺ gz z ∈ U or f ≺ g in U.
1.3
2
International Journal of Mathematics and Mathematical Sciences
It is well known 2 that if g is univalent in U, then f ≺ g in U is equivalent to f0 g0 and
fU ⊂ gU.
For the functions f and g given by the power series
fz ∞
an zn ,
gz n0
∞
bn z n
z ∈ U,
1.4
n0
their Hadamard product or convolution, denoted by f∗g, is defined by
f∗gz ∞
an bn zn g∗fz z ∈ U.
1.5
n0
Note that f∗g ∈ A.
By making use of the Hadamard product, Carlson-Shaffer 3 defined the linear operator
La, c : A → A by
La, cf z : Φa, c; z∗fz z ∈ U, f ∈ A,
1.6
where
Φa, c; z :
∞
ak
k0
ck
zk1
z ∈ U, c/
∈Z−0 {0} ∪ {−1, −2, −3, . . .}
1.7
and λk is the Pochhammer symbol or shifted factorial defined in terms of the gamma function
by
Γλ k
λk Γλ
k 0,
λλ 1λ 2 · · · λ k − 1 k ∈ N : {1, 2, . . .} .
1
1.8
It can be readily verified that La, a a /
∈ Z−0 is the identity operator; the operators
−
La, b, Lc, d commute, where b, d/
∈Z0 , that is,
La, bLc, df Lc, dLa, bf
f ∈ A,
1.9
and the transitive property, that is,
La, bLb, cf La, cf
b, c /
∈ Z−0 , f ∈ A ,
1.10
holds. Each of the following definitions will also be required in our present investigation.
Definition 1.1 cf. 4, 5, see also 6. Let the function f be analytic in a simply connected region
of the z-plane containing the origin. The fractional derivative of f of order λ is defined by
Dzλ f z d
1
Γ1 − λ dz
z
0
fζ
z − ζλ
dζ
0 ≤ λ < 1,
1.11
where the multiplicity of z − ζλ is removed by requiring logz − ζ to be real when z − ζ > 0.
A. K. Mishra and P. Gochhayat
3
Using Definition 1.1 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava 5 introduced the fractional differintegral operator Ωλz :
A0 → A0 defined by
λ 1.12
Ωz f z Γ2 − λzλ Dzλ f z λ /
2, 3, . . . , z ∈ U.
Note that Ω0z fz fz, Ω1z fz zf z, and
λ Ωz f z L2, 2 − λf z 0 ≤ λ < 1, z ∈ U.
1.13
Definition 1.2 cf. 7. For the function f given by 1.2 and q ∈ N : {1, 2, 3, . . .}, the qth Hankel
determinant of f is defined by
a
an1 · · · anq−1 n
an1 an2 · · · anq .
.
1.14
..
..
..
.
.
.
.
.
anq−1 anq · · · an2q−2 We now introduce the following class of functions.
Definition 1.3. The function f ∈ A0 is said to be in the class Rλ α, ρ 0 ≤ λ < 1, |α| < π/2, 0 ≤
ρ ≤ 1 if it satisfies the inequality
λ
iα Ωz fz
> ρ cos α z ∈ U.
1.15
R e
z
Write
Rλ 0, ρ : Rλ ρ.
1.16
Let P be the family of functions p ∈ A satisfying p0 1 and Rpz > 0 z ∈ U.
It follows from 1.15 that
f ∈ Rλ α, ρ ⇐⇒ eiα
Ωλz fz 1 − ρpz ρ cos α i sin α,
z
where α is real, |α| < π/2, and pz ∈ P.
We note that
iα fz
> ρ cos α ,
R0 α, ρ : f ∈ A0 | R e
z
R1 α, ρ : f ∈ A0 | R eiα f z > ρ cos α ,
1.17
1.18
and the class Rλ ρ has been studied in 8.
It is well known cf. 2 that for f ∈ S and given by 1.2, the sharp inequality |a3 − a22 | ≤
1 holds. This corresponds to the Hankel determinant with q 2 and n 1. For a given family
F of functions in A0 , the more general problem of finding sharp estimates for |μa22 − a3 | μ ∈
R or μ ∈ C is popularly known as the Fekete-Szegö problem for F. The Fekete-Szegö problem
for the families S, S ∗ , CV, K has been completely solved by many authors including 9–12.
In the present paper, we consider the Hankel determinant for q 2 and n 2 and we
find the sharp bound for the functional |a2 a4 − a23 | f ∈ Rλ α, ρ. We also obtain some basic
properties of the class Rλ α, ρ. Our investigation includes a recent result of Janteng et al. 13.
We also generalize some results of Ling and Ding 8.
4
International Journal of Mathematics and Mathematical Sciences
2. Preliminaries
To establish our results, we recall the following.
Lemma 2.1 see 2. Let the function p ∈ P and be given by the series
pz 1 c1 z c2 z2 · · ·
z ∈ U.
2.1
Then, the sharp estimate
ck ≤ 2 k ∈ N
2.2
holds.
Lemma 2.2 cf. 14, page 254, see also 15. Let the function p ∈ P be given by the power series
2.1. Then,
2c2 c12 x 4 − c12
2.3
for some x, |x| ≤ 1, and
4c3 c13 24 − c12 c1 x − c1 4 − c12 x2 24 − c12 1 − |x|2 z
2.4
for some z, |z| ≤ 1.
Lemma 2.3 see 16. Let F and G be univalent convex functions in U. Then, the Hadamard product
F∗G is also a univalent convex function in U.
Lemma 2.4 see 17. Let F and G be univalent convex functions in U. Also let f ≺ F and g ≺ G in
U. Then, f∗g ≺ F∗G in U.
Lemma 2.5 see 16, also see 8. Let f and g be starlike of order 1/2. Then, for each function Fz,
satisfying RFz > α 0 ≤ α < 1, z ∈ U, one has
fz∗Fzgz
> α z ∈ U.
2.5
R
fz∗gz
Lemma 2.6 see 8. Let the function hz 1 h1 z h2 z2 · · · be univalent convex in U. For
0 ≤ λ < 1 if Ωλz fz/z ≺ hz z ∈ U, then
fz ≺ L2 − λ, 2 zhz
z
z ∈ U.
2.6
3. Main results
We prove the following.
Theorem 3.1. Let the function f given by 1.2 be in the class Rλ α, ρ 0 ≤ λ < 1, − π/2 < α <
π/2, and 0 ≤ ρ ≤ 1. Then,
2
2
2
2
a2 a4 − a2 ≤ 1 − ρ 2 − λ 3 − λ cos α .
3
9
The estimate 3.1 is sharp.
3.1
A. K. Mishra and P. Gochhayat
5
Proof. Let f ∈ Rλ α, ρ 0 ≤ λ < 1, − π/2 < α < π/2, and 0 ≤ ρ ≤ 1. Then, by 1.17,
eiα
Ωλz fz 1 − ρpz ρ cos α i sin α z ∈ U,
z
3.2
where p ∈ P and is given by 2.1. Using 1.6, 1.7, and 1.13, we write
Ωλz fz z ∞
Γn 1Γ2 − λ
n2
Γn 1 − λ
an zn ,
z ∈ U.
3.3
Comparing the coefficients, we get
eiα
eiα
eiα
2
a2 1 − ρc1 cos α,
2 − λ
6
a3 1 − ρc2 cos α,
2 − λ3 − λ
3.4
24
a4 1 − ρc3 cos α.
2 − λ3 − λ4 − λ
Therefore, 3.4 yields
2
2
2
3 − λc22 a2 a4 − a2 1 − ρ 2 − λ 3 − λcos α 4 − λc1 c3 −
.
3
12
4
3
3.5
Since the functions pz and peiθ z, θ ∈ R are members of the class P simultaneously, we
assume without loss of generality that c1 > 0. For convenience of notation, we take c1 c c ∈
0, 2.
Using 2.3 along with 2.4, we get
2
2
2
a2 a4 − a2 1 − ρ 2 − λ 3 − λcos α
3
12
4 − λc 3
× c 2 4 − c2 cx − c 4 − c2 x2 24 − c2 1 − |x|2 z 16
1 − ρ2 2 − λ2 3 − λcos2 α
48
4 − λ 3 − λ 4
4 − λ 4 − c2 c2 2c2 3 − λ 4 − c2
−
c −
x
× 4
3
2
3
2 4 − λ 4 − c2 c2 3 − λ 4 − c2
4 − λ 4 − c2 c 1 − |x|2 z 2
−
x 4
3
2
1 − ρ2 2 − λ2 3 − λ cos2 α
48
4
λc λ 4 − c2 c2 x
48 − λ 16 − c2 2 4 − λ 4 − c2 c 1 − |x|2 z 2
.
−
×
4−c x 12
6
12
2
3.6
6
International Journal of Mathematics and Mathematical Sciences
An application of triangle inequality and replacement of |x| by μ give
1 − ρ2 2 − λ2 3 − λ cos2 α
2
a2 a4 − a ≤
3
48
4
λ 4 − c2 c2 μ
4 − c2 48 − λ 16 − c2 μ2 4 − λ 4 − c2 c
λc
×
12
6
12
2
2
2
4 − λ 4 − c cμ
−
2
1 − ρ2 2 − λ2 3 − λ cos2 α
48
4
4 − λ 4 − c2 c λ 4 − c2 c2 μ
λc
×
12
2
6
2
λ c − 64 − λ c/λ 163 − λ /λ 4 − c2 μ2
12
2 2
2
1 − ρ 2 − λ 3 − λ cos α
48
4
4 − λ 4 − c2 c λ 4 − c2 c2 μ λ c − β1 c − β2 4 − c2 μ2
λc
×
12
2
6
12
3.7
: Fc, μ say,
where
β1 2,
β2 83 − λ
,
λ
0 ≤ c ≤ 2,
0 ≤ μ ≤ 1.
We next maximize the function Fc, μ on the closed square 0, 2 × 0, 1. Since
2
2
∂F 1 − ρ 2 − λ 3 − λcos2 α λ 4 − c2 c2 λ 4 − c2 c − 2 c − 83 − λ/λ μ
,
∂μ
48
6
6
3.8
3.9
c − 2 < 0, and c − 83 − λ/λ < 0, we have ∂F/∂μ > 0 for 0 < c < 2, 0 < μ < 1. Thus Fc, μ
cannot have a maximum in the interior of the closed square 0, 2 × 0, 1. Moreover, for fixed
c ∈ 0, 2,
max Fc, μ Fc, 1 Gc say.
3.10
−1 − ρ2 2 − λ2 3 − λ c2 − 7λ − 12 c cos2 α
G c ,
72
3.11
0≤μ≤1
Next,
so that G
c < 0 for 0 < c < 2 and has real critical point at c 0. Also Gc > G2. Therefore,
max0≤c≤2 occurs at c 0. Therefore, the upper bound of 3.7 corresponds to μ 1 and c 0.
Hence,
2
2
2
2
a2 a4 − a2 ≤ 1 − ρ 2 − λ 3 − λ cos α
3
9
3.12
A. K. Mishra and P. Gochhayat
7
which is the assertion 3.1. Equality holds for the function
1 1 − 2ρz2
fz Φ2 − λ, 2; z∗e−iα z
cos
α
i
sin
α
.
1 − z2
3.13
The proof of Theorem 3.1 is complete.
The choice of α 0 yields what follows.
Corollary 3.2. Let the function f given by 1.2 be a member of the class Rλ ρ. Then,
2
2
2
a2 a4 − a2 ≤ 1 − ρ 2 − λ 3 − λ .
3
9
3.14
Equality holds for the function
z 1 1 − 2ρz2
.
fz L2 − λ, 2∗
1 − z2
3.15
Remark 3.3. Taking λ → 1, α 0, and ρ 0, we get a recent result due to Janteng et al. 13.
Theorem 3.4. Suppose −π/2 < α < π/2, 0 ≤ ρ < 1, and 0 ≤ μ < λ < 1. Then,
Rλ α, ρ ⊂ Rμ α, ρ.
3.16
Proof. Let
f ∈ Rλ α, ρ
π
π
0 ≤ μ < λ < 1, − < α < , 0 ≤ ρ ≤ 1 .
2
2
3.17
Using the associative and commutative properties of the operator L, we write
μ
Ωz fz L2, 2 − μfz
L2 − λ, 2L2, 2 − λL2, 2 − μfz
L2 − λ, 2 − μΩλz fz
3.18
Φ2 − λ, 2 − μ; z∗Ωλz fz,
where the function Φ is defined by 1.7. Therefore,
μ
eiα Ωz fz Φ2 − λ, 2 − μ; z∗ eiα Ωλz fz/z ·z
z
Φ2 − λ, 2 − μ; z∗z
fz∗Fzgz
,
fzgz
3.19
where fz Φ2 − λ, 2 − μ; z, gz z, Fz eiα Ωλz fz/z. We note that g ∈ S ∗ 1/2, and
RFz > ρ cos α 0 ≤ ρ ≤ 1, − π/2 < α < π/2. Moreover, it is well known cf. 18 that
8
International Journal of Mathematics and Mathematical Sciences
Φ2 − λ, 2 − μ; z ∈ S ∗ 1/2. Therefore, by Lemma 2.5,
R
μ
eiα Ωz fz
z
> ρ cos α
−
π
π
< α < , z ∈ U, 0 ≤ ρ ≤ 1 .
2
2
3.20
Hence, fz ∈ Rμ α, ρ, and the proof of Theorem 3.4 is complete.
Theorem 3.5. Let f ∈ S ∗ 1/2 and g ∈ Rλ α, ρ 0 ≤ ρ ≤ 1, − π/2 < α < π/2, 0 ≤ λ < 1. Then
the Hadamard product
f∗g ∈ Rλ α, ρ.
3.21
Proof. Since the Hadamard product is associative and commutative, we have
Ωλz f∗gz fz∗Ωλz gz.
3.22
eiα Ωλz f∗gz fz∗ eiα Ωλz gz/z ·z
.
z
fz∗z
3.23
Therefore,
Now applying Lemma 2.5, we get
R
eiα Ωλz f∗gz
z
> ρ cos α.
3.24
Hence, f∗g ∈ Rλ α, ρ, and the proof of Theorem 3.5 is complete.
Theorem 3.6. Let f ∈ Rλ α, ρ 0 ≤ λ < 1, − π/2 < α < π/2, 0 ≤ ρ ≤ 1. Then, the function If
defined by the integral transform
γ 1 z γ−1
Ifz γ
t ftdt z ∈ U, γ > −1
3.25
z
0
is also in Rλ α, ρ.
Proof. The Integral transform If can be written in terms of Carlson-Shaffer operator as
If z Lγ 1, γ 2f z.
3.26
Ωλz If z Lγ 1, γ 2Ωλz fz Φγ 1, γ 2; z∗Ωλz fz.
3.27
eiα Ωλz If z Φγ 1, γ 2; z∗ eiα Ωλz fz/z z
.
z
Φγ 1, γ 2; z∗z
3.28
Hence,
Therefore,
Using a result of Bernardi 19, it can be verified that Φγ 1, γ 2; z ∈ S ∗ 1/2. Thus by
applying Lemma 2.5, the proof of Theorem 3.6 is complete.
A. K. Mishra and P. Gochhayat
9
Theorem 3.7. Let f ∈ Rλ α, ρ, 0 ≤ λ < 1, − π/2 < α < π/2, 0 ≤ ρ ≤ 1. Then,
fz
≺ Gz
z
z ∈ U,
3.29
where
e−iα Φ2 − λ, 2; z∗ zhz ,
Gz z
1 1 − 2ρz
hz cos α i sin α ,
1−z
3.30
and Φ is defined by 1.7. Moreover, G is a univalent convex function in U.
Proof. Since Ωλz fz/z ≺ e−iα hz, by an application of Lemma 2.6, we get
fz e−iα ≺
L2 − λ, 2∗ zhz Gz.
z
z
3.31
The assertion 3.29 is proved.
It is well known cf. 18 that Φ2 − λ, 2; z/z is a univalent convex function. Therefore,
by Lemma 2.4, Gz is univalent convex function.
Remark 3.8. For α 0, Theorem 3.7i gives a result of Ling and Ding 8, Theorem 2.
Acknowledgments
The present investigation is partially supported by National Board for Higher Mathematics,
Department of Atomic Energy, Government of India under Grant no. 48/2/2003-R&D-II.
References
1 M. I. S. Robertson, “On the theory of univalent functions,” Annals of Mathematics, vol. 37, no. 2, pp.
374–408, 1936.
2 P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer,
New York, NY, USA, 1983.
3 B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on
Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984.
4 S. Owa, “On the distortion theorems. I,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59,
1978.
5 S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canadian Journal of Mathematics, vol. 39, no. 5, pp. 1057–1077, 1987.
6 H. M. Srivastava and S. Owa, “An application of the fractional derivative,” Mathematica Japonica,
vol. 29, no. 3, pp. 383–389, 1984.
7 J. W. Noonan and D. K. Thomas, “On the second Hankel determinant of areally mean p-valent functions,” Transactions of the American Mathematical Society, vol. 223, no. 2, pp. 337–346, 1976.
8 Y. Ling and S. S. Ding, “A class of analytic functions defined by fractional derivation,” Journal of Mathematical Analysis and Applications, vol. 186, no. 2, pp. 504–513, 1994.
9 M. Fekete and G. Szegö, “Eine Bemerkung über ungerade schlichte Funktionen,” Journal of the London
Mathematical Society, vol. 8, pp. 85–89, 1933.
10 F. R. Keogh and E. P. Merkes, “A coefficient inequality for certain classes of analytic functions,” Proceedings of the American Mathematical Society, vol. 20, pp. 8–12, 1969.
10
International Journal of Mathematics and Mathematical Sciences
11 W. Koepf, “On the Fekete-Szegö problem for close-to-convex functions,” Proceedings of the American
Mathematical Society, vol. 101, no. 1, pp. 89–95, 1987.
12 W. Koepf, “On the Fekete-Szegö problem for close-to-convex functions—II,” Archiv der Mathematik,
vol. 49, no. 5, pp. 420–433, 1987.
13 A. Janteng, S. A. Halim, and M. Darus, “Coefficient inequality for a function whose derivative has
positive real part,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 2, p. 50, 2006.
14 R. J. Libera and E. J. Złotkiewicz, “Coefficient bounds for the inverse of a function with derivative in
P,” Proceedings of the American Mathematical Society, vol. 87, no. 2, pp. 251–257, 1983.
15 R. J. Libera and E. J. Złotkiewicz, “Early coefficients of the inverse of a regular convex function,”
Proceedings of the American Mathematical Society, vol. 85, no. 2, pp. 225–230, 1982.
16 St. Ruscheweyh and T. Sheil-Small, “Hadamard products of Schlicht functions and the PólyaSchoenberg conjecture,” Commentarii Mathematici Helvetici, vol. 48, pp. 119–135, 1973.
17 St. Ruscheweyh and J. Stankiewicz, “Subordination under convex univalent functions,” Bulletin of the
Polish Academy of Sciences. Mathematics, vol. 33, no. 9-10, pp. 499–502, 1985.
18 D. G. Yang, “Subclasses of starlike functions of order α,” Chinese Annals of Mathematics. Series A, vol. 8,
no. 6, pp. 687–692, 1987.
19 S. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical
Society, vol. 135, pp. 429–446, 1969.