Inclusion Properties of Hypergeometric Type Functions and Related Integral Transforms (Research Paper Sample)

Inclusion properties of hypergeometric type functions and related integral transforms
Abstract. In this work, conditions on the parameters a, b and c are given so that the normalized Gaussian hypergeometric function zF (a, b; c; z), where
Σ (a) (b)nn n| |∞
F (a, b; c; z) =z ,z < 1, (c)n(1)n
n=0
is in certain class of analytic functions. Using Taylor coefficients of functions in certain classes, inclusion properties of the Hohlov integral transform involving zF (a, b; c; z) are obtained. Similar inclusion results of the Komatu integral oper- ator related to the generalized polylogarithm are also obtained. Various results for the particular values of these parameters are deduced and compared with the existing literature.
Mathematics Subject Classification (2010): 30C45, 33C45, 33A30.
Keywords: Univalent, convex, starlike, close-to-convex functions, Gaussian hyper- geometric functions, incomplete beta functions, Komatu integral operator, poly- logarithm.
1 Introduction
ΣkLet A denote the class of functions of the form
∞
f (z) = z +akz ,(1.1)
k=2
g ∈ S∗ if and only if Reeiλ zfj(z)g(z)> 0 for z ∈ D and λ ∈ R. . Let K denote theanalytic in the open unit disk D = {z : |z| < 1}, and S denote the subclass of A that contains functions univalent in D. A function f ∈ A is called starlike, denoted by f ∈ S∗, if tw ∈ f (D) whenever w ∈ f (D) and t ∈ [0, 1]. The class of all convex functions, denoted by C, consists of the functions f ∈ A such that zfj is starlike. A function f ∈ A is said to.be close-toΣ-convex with respect to a fixed starlike function
subclass of all such close-to-convex functions, where λ = 0. Various generalization of these classes and various other subclasses of S exist in the literature. For example the class of starlike functions of order σ, denoted by S∗(σ), 0 ≤ σ < 1, which has the analytic characterization Re zfj(z) > σ, is the generalization of the class S∗(0) = S∗.
f (z)
Note that C(σ), the class of convex functions of order σ contains all functions f ∈ S
for which zfj ∈ S∗(σ).
γ,αWe introduce the class Rτ (β), with 0 ≤ γ < 1, 0 ≤ α ≤ 1, τ ∈ C\{0} and
β < 1 as
. 2τ (1 − β) + (1 − α + 2γ) f + (α − 2γ)fj + γzfjj − 1 ...(1 − α + 2γ) f + (α − 2γ)fj + γzfjj − 1.Σ
γ,αz
z
(1.2)
Rτ (β) :=
f ∈ A :.
. < 1, z ∈ D .
Note that few particular cases of this class discussed in the literature.
1 γ,αThe class Rτ
(β) for α = 2γ + 1, was considered in [16], where references about
other particular cases in this direction are provided.
2 γ, αThe class Rτ(β) for τ = eiη cos η, where −π/2 < η < π/2 is considered in [1]
(see also [2, 3]), and the properties of certain integral transforms of the type
| |Vλ(f ) =0λ(t)tdt,f ∈ R0,γ(β)(1.3)∫ 1f (tz)
(eiη cos η)
with β < 1, γ < 1 and η < π/2, under suitable restriction on λ(t) was discussed using duality techniques for various values of γ in [1]. For other interesting cases, we refer to [3, 16] and references therein.
3 0,1The class Rτ (0) with τ = eiη cos η was considered in [10] with reference to the
univalency of partial sums.
γ, αIt is clear that the geometric properties of certain integral transforms under du- ality techniques, which is one of recent research interest (for example, see [1, 3] and references therein), cannot be proved easily as the results involve certain multiple integrals and it is difficult to check the conditions given for the existence of the inclu- sion results for these integral transforms. For this purpose, the inclusion properties of certain special functions to be in the analytic subclasses like R(eiη cos η)(β) are studied using techniques other than duality methods which motivates this work.
Among various results related to the integral operator (1.3) available in the
literature, an important and interesting result is application of the operator (1.3) when λ(t) is related to the function zF (a, b; c; z). Here by F (a, b; c; z) we mean the well-known Gaussian hypergeometric function
∞
Σ (a)n(b)n nF (a, b; c; z) =z(1.4)
n=0 (c)n(1)n
∈z D, with (λ)n being the Pochhammer symbol given by (λ)n = λ(λ+1)n−1, (λ)0 = 1. Also, there has been considerable interest to find conditions on the parameters a, b, and c such that the normalized hypergeometric functions (c/ab) (F (a, b; c; z) − 1) or zF (a, b; c; z) belong to one of the known subclasses of S. For more details on the basic
∗∗ideas of Gaussian hypergeometric functions, we refer to [11] and on the applications related to geometric function theory, we refer to [1, 14, 15, 16] and references therein. Related to F (a, b; c; z) is the Hohlov operator Ha, b, c (f )(z) = zF (a, b; c; z) f (z), wheredenotes the well-known Hadamard product or convolution. This operator is particular case of a general integral transform studied in [5]. To be more specific, the
∫properties of certain integral transforms of the type
Vλ(f ) =
1
λ(t)
f (tz)
t
dt,f ∈ R(e
cos η)(β)(1.5)
iηγ, α
0
under suitable restriction on λ(t) was discussed by many authors [1, 3, 5]. In particular,
if
−λ(t) = Γ(c)tb−1(1t)c−b−1
Γ(b)Γ(c − b)
Lthen Vλ(f ) = L(b, c)(f )(z) which is the well-known Carlson-Schaffer operator. Note that H 1, b, c (f )(z) = (b, c)(f )(z). The following lemma exhibits the relation between the integral operator in discussion with the Hohlov operator.
Lemma 1.1. If f ∈ A and c − a + 1 > b > 0, then
Vλ(f )(z) = Ha,b,c(f )(z)
where
Ha,b,c(f )(z) =
Γ(c) ∫ 1
(1 − t)c−a−b
tb−2F (c−a, 1−a; c−a−b+1; 1−t)f (tz)dt.
Γ(a)Γ(b) 0 Γ(c − a − b + 1)
aThe Komatu operator Kp : A → A [9] is defined as
Kp[f ](z) =
(1 + a)p ∫ 1 .
1
log(
)Σp−1
ta−1f (tz)dt,
aΓ(p)0t
where a > −1 and p ≥ 0. It has a series representation as
an=2(n + a)pn∞p
Kp[f ](z) = z + Σ (1 + a) a zn
and in terms of convolution, we can write
Kp[f ](z) = Kp(z) ∗ f (z),
aa
a(n + a)p∞p
where Kp(z) = z + Σ (1 + a) zn.
an=2In this paper we study the operators Ha, b, c (f )(z) and Kp[f ](z) for various
choices of the function f .
The paper is organized as follows. In Section 2, some preliminary results about the Gaussian hypergeometric function F (a, b; c; z) and conditions on the Taylor co-
γ,αefficients of f ∈ Rτ (β) are given which are used in the subsequent sections. Con-
ditions on the triplets a, b, c are obtained so that in Section 3 inclusion properties of
γ,α∗F (a, b; c; z) and its normalized case to be in the class Rτ
(β) are discussed and in
Section 4, inclusion properties of zF (a, b; c; z) f (z) for f in various subclasses of S
are discussed. Similar type of inclusion results for the Komatu operator is discussed
in Section 5. In the last section, certain remarks are given to provide motivation for further research in this direction.
2 Preliminary results
The following result is available in [16], which can also be easily verified by simple computation.
Lemma 2.1. Let F (a, b; c; z) be the Gaussian hypergeometric function as given in (1.4).
Then we have the following
(i) For Re (c − a − b) > 0 and c ƒ= 0, −1, −2, . . .,
F (a, b; c; 1) = Γ(c − a − b)Γ(c)
Γ(c − a)Γ(c − b)
(2.1)
* For a, b > 0, c > a + b + 1,
∞n=0(c)n(1)nc − 1 − a − bΣ (n + 1)(a)n(b)n = F (a, b; c; 1) Σ ab+ 1Σ .(2.2)
* ∞n=0 (c)n(1)n+1(a − 1)(b − 1)For a ƒ= 1, b ƒ= 1 and c ƒ= 1 with c > max{0, a + b − 1},
Σ (a)n(b)n = (c − 1)ΣF (a − 1, b − 1; c − 1; 1) − 1Σ.(2.3)
* For a ƒ= 1 and c ƒ= 1 with c > max{0, 2Re a − 1},
∞n=0(c)n(1)n+1= |a − 1|2F (a − 1, a¯ − 1; c − 1; 1) − 1.(2.4)Σ|(a)n|2 (c − 1) ΣΣ
Proof. Part (i) is the ...