Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains
Let A be the class of functions f(z) = z + ∑ k = 2∞ a k z k analytic in an open unit disc ∆. We use a generalized linear operator closely related to the multiplier transformation to study certain subclasses of A mapping ∆ onto conic domains. Using the principle of the differential subordination and...
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irk-123456789-1658612020-02-17T01:27:32Z Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains Deniz, E. Orhan, H. Sokół, J. Статті Let A be the class of functions f(z) = z + ∑ k = 2∞ a k z k analytic in an open unit disc ∆. We use a generalized linear operator closely related to the multiplier transformation to study certain subclasses of A mapping ∆ onto conic domains. Using the principle of the differential subordination and the techniques of convolution, we investigate several properties of these classes, including some inclusion relations and convolution and coefficient bounds. In particular, we get many known and new results as special cases. Нехай A — клас функцій f(z) = z + ∑∞k = 2akzk, аналітичних у відкритому одиничному крузі Δ. До вивчення деяких підкласів A, що відображають Δ на конічні області, застосовано узагальнений лінійний оператор, тісно пов'язаний з перетворенням множення. За допомогою принципу диференціального підпорядкування та техніки згорток вивчено деякі властивості цих класів, що включають деякі співвідношення включення та згорток, а також оцінки для коефіцієнтів. Наприклад, низку відомих та нових результатів отримано як частинні випадки. 2015 Article Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains / E. Deniz, H. Orhan, J. Sokół // Український математичний журнал. — 2015. — Т. 67, № 9. — С. 1217–1231. — Бібліогр.: 34 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165861 517.54 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Deniz, E. Orhan, H. Sokół, J. Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains Український математичний журнал |
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Let A be the class of functions f(z) = z + ∑ k = 2∞ a k z k analytic in an open unit disc ∆. We use a generalized linear operator closely related to the multiplier transformation to study certain subclasses of A mapping ∆ onto conic domains. Using the principle of the differential subordination and the techniques of convolution, we investigate several properties of these classes, including some inclusion relations and convolution and coefficient bounds. In particular, we get many known and new results as special cases. |
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Deniz, E. Orhan, H. Sokół, J. |
| author_facet |
Deniz, E. Orhan, H. Sokół, J. |
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Deniz, E. |
| title |
Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains |
| title_short |
Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains |
| title_full |
Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains |
| title_fullStr |
Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains |
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Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains |
| title_sort |
classes of analytic functions defined by a differential operator related to conic domains |
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Інститут математики НАН України |
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2015 |
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Статті |
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| citation_txt |
Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains / E. Deniz, H. Orhan, J. Sokół // Український математичний журнал. — 2015. — Т. 67, № 9. — С. 1217–1231. — Бібліогр.: 34 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT denize classesofanalyticfunctionsdefinedbyadifferentialoperatorrelatedtoconicdomains AT orhanh classesofanalyticfunctionsdefinedbyadifferentialoperatorrelatedtoconicdomains AT sokołj classesofanalyticfunctionsdefinedbyadifferentialoperatorrelatedtoconicdomains |
| first_indexed |
2025-07-14T20:09:59Z |
| last_indexed |
2025-07-14T20:09:59Z |
| _version_ |
1837654408399159296 |
| fulltext |
UDC 517.54
E. Deniz (Kafkas Univ., Kars, Turkey),
H. Orhan (Ataturk Univ., Erzurum, Turkey),
J. Sokół (Rzeszów Univ. Technology, Poland)
CLASSES OF ANALYTIC FUNCTIONS DEFINED
BY A DIFFERENTIAL OPERATOR RELATED TO CONIC DOMAINS*
КЛАСИ АНАЛIТИЧНИХ ФУНКЦIЙ,
ЩО ВИЗНАЧЕНI ДИФЕРЕНЦIАЛЬНИМ ОПЕРАТОРОМ,
ВIДНЕСЕНИМ ДО КОНIЧНИХ ОБЛАСТЕЙ
Let A be the class of functions f(z) = z +
∑∞
k=2
akz
k, which are analytic in the open unit disc ∆. We use a generalized
linear operator closely related to the multiplier transformation to investigate certain subclasses of A which map ∆ onto the
conic domains. Using the principle of the differential subordination and the techniques of convolution, several properties
of these classes including some inclusion relations, convolution and coefficient bounds are studied. In particular, we derive
many known and new results as special cases.
Нехай A — клас функцiй f(z) = z +
∑∞
k=2 akz
k, аналiтичних у вiдкритому одиничному крузi ∆. До вивчення
деяких пiдкласiв A, що вiдображають ∆ на конiчнi областi, застосовано узагальнений лiнiйний оператор, тiсно
пов’язаний з перетворенням множення. За допомогою принципу диференцiального пiдпорядкування та технiки
згорток вивчено деякi властивостi цих класiв, що включають деякi спiввiдношення включення та згорток, а також
оцiнки для коефiцiєнтiв. Наприклад, низку вiдомих та нових результатiв отримано як частиннi випадки.
1. Introduction. Let A be the class of functions of the form
f(z) = z +
∞∑
k=2
akz
k, (1.1)
analytic in the open unit disk ∆ = {z : z ∈ C and |z| < 1}. Let S denote the class of functions
f ∈ A which are univalent in ∆. If f and g are analytic in ∆, we say that f is subordinate to g,
written symbolically as f ≺ g or f(z) ≺ g(z) if there exists a Schwarz function w(z), is analytic
in ∆ (with w(0) = 0 and |w(z)| < 1 in ∆) such that f(z) = g(w(z)), z ∈ ∆. In particular, if the
function g(z) is univalent in ∆, then f(z) ≺ g(z) if and only if f(0) = g(0) and f(∆) ⊆ g(∆).
A function f ∈ A is said to be in the class of uniformly convex functions of order γ and type β,
denoted by β − UCV(γ) [5] if
<
{
1 +
zf ′′(z)
f ′(z)
}
> β
∣∣∣∣zf ′′(z)f ′(z)
∣∣∣∣+ γ, z ∈ ∆, (1.2)
where β ≥ 0, −1 ≤ γ < 1, β + γ ≥ 0 and it is said to be in the corresponding class denoted by
β − SP(γ) if
<
{
zf ′(z)
f(z)
}
> β
∣∣∣∣zf ′(z)f(z)
− 1
∣∣∣∣+ γ, z ∈ ∆, (1.3)
where β ≥ 0, −1 ≤ γ < 1 and β + γ ≥ 0.
* This paper was supported by the Commission for the Scientific Research Projects of Kafkas Univertsity (Project
number 2012-FEF-30).
c© E. DENIZ, H. ORHAN, J. SOKÓL, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9 1217
1218 E. DENIZ, H. ORHAN, J. SOKÓL
These classes generalize various other classes which are worthy to mention here. For example
the class β − UCV(0) = β − UCV is the known class of β-uniformly convex functions [11]. Using
the Alexander type relation, we can obtain the class β − SP(γ) in the following way:
f ∈ β − SP(γ)⇔ 1
z
z∫
0
f(t)dt ∈ β − UCV(γ) or f ∈ β − UCV(γ)⇔ zf ′ ∈ β − SP(γ).
The class 1−UCV(0) = UCV of uniformly convex functions was defined by Goodman [9] while
the class 1− SP(0) = SP was considered by Rønning [26].
Geometric interpretation. It is known that f ∈ β − UCV(γ) and g ∈ β − SP(γ) if and only
if the quantities 1 + zf ′′(z)/f ′(z) and zg′(z)/g(z), respectively, takes its all the values in the conic
domain Rβ,γ which is included in the right half plane <(w) > (β + γ)/(1 + β) and is given by
Rβ,γ :=
{
w = u+ iv ∈ C : u > β
√
(u− 1)2 + v2 + γ, β ≥ 0 and γ ∈ [−1, 1)
}
. (1.4)
Let P̂β,γ = 1 + P1z + . . . denote the function which maps the unit disk conformally onto the
domain Rβ,γ given in (1.4). Let ∂Rβ,γ be a curve defined by the equality
∂Rβ,γ :=
{
w = u+ iv ∈ C : u2 =
(
β
√
(u− 1)2 + v2 + γ
)2
, β ≥ 0 and γ ∈ [−1, 1)
}
. (1.5)
After some calculations we can see that for β 6= 0, ∂Rβ,γ represents conic curves symmetric
about the real axis. Thus Rβ,γ is an elliptic domain for β > 1, a parabolic domain for β = 1, a
hyperbolic domain for 0 < β < 1 and the right half plane <(w) > γ, for β = 0.
The functions P̂β,γ play the role of extremal functions in the classes P(P̂β,γ) and were given in
[1] (also see for Taylor series expansion of P̂β,γ , [14, 16, 26]) as follows:
P̂β,γ(z) =
1 + (1− 2γ)z
1− z
, β = 0,
1 +
2(1− γ)
π2
(
log
1 +
√
z
1−
√
z
)2
, β = 1,
1− γ
1− β2
cos
{
2
π
(arccosβ)i log
1 +
√
z
1−
√
z
}
− β2 − γ
1− β2
, 0 < β < 1,
1− γ
β2 − 1
sin
{
π
2K(t)
∫ u(z)√
t
0
dx√
1− x2
√
1− t2x2
}
+
β2 − γ
β2 − 1
, β > 1,
(1.6)
where u(z) =
z −
√
t
1−
√
tz
, t ∈ (0, 1), z ∈ ∆ and
K(t) =
1∫
0
dx√
1− x2
√
1− t2x2
(1.7)
is called Legendre’s complete elliptic integral of the first kind and t ∈ (0, 1) is such that β =
= coshπK′(t)/4 K(t).
For functions f, g ∈ A, given by f(z) = z +
∑∞
k=2
akz
k and g(z) = z +
∑∞
k=2
bkz
k, the
Hadamard product (or convolution) of f and g is defined by
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
CLASSES OF ANALYTIC FUNCTIONS DEFINED BY A DIFFERENTIAL OPERATOR . . . 1219
(f ∗ g)(z) := z +
∞∑
k=2
akbkz
k, z ∈ ∆.
Note that f ∗ g ∈ A. Let a ∈ R, c ∈ R, c 6= 0,−1,−2, . . . and let
ϕ(a, c; z) := z +
∞∑
k=1
(a)k
(c)k
zk+1, z ∈ ∆, (1.8)
where (κ)n is the Pochhammer symbol (or the shifted factorial) in terms of the gamma function,
given by
(κ)n :=
Γ(κ+ n)
Γ(κ)
=
1, n = 0, κ ∈ C \ {0} ,
κ(κ+ 1) . . . (κ+ n− 1), n ∈ N = {1, 2, . . .}, κ ∈ C.
The Carlson – Shaffer operator [6] L(a, c) is defined in terms of Hadamard product by
L(a, c)f(z) = ϕ(a, c; z) ∗ f(z), z ∈ ∆, f ∈ A. (1.9)
Note that L(a, a) is the identity operator and L(a, c) = L(a, b)L(b, c), (b, c 6= 0,−1,−2, . . .). We
also need the following definitions of a fractional derivative.
Definition 1.1 [21]. Let the function f be analytic in a simply-connected region of the z-plane
containing the origin. The fractional derivative of fof order α is defined by
Dα
z f(z) :=
1
Γ(1− α)
d
dz
z∫
0
f(ζ)
(z − ζ)α
dζ, 0 ≤ α < 1,
where the multiplicity of (z − ζ)−α is removed by requiring log(z − ζ) to be real when z − ζ > 0.
Using Dα
z Owa and Srivastava [22] introduced the operator Ωα : A → A, α ∈ [0, 1) , which is
known as an extension of fractional derivative and fractional integral, as follows:
Ωαf(z) = Γ(2− α)zαDα
z f(z) = z +
∞∑
k=2
Γ(k + 1)Γ(2− α)
Γ(k + 1− α)
akz
k =
= ϕ(2, 2− α; z) ∗ f(z) = L(2, 2− α)f(z). (1.10)
Note that Ω0
zf(z) = f(z).
In [20], Orhan, Deniz and Răducanu introduced the generalized linear multiplier fractional
differential operator Dn,α
λ,µf : A → A of functions f ∈ A defined by
D0,α
λ,µf(z) = f(z),
D1,α
λ,µf(z) = Dα
λ,µf(z) = λµz2[Ωαf(z)]′′ + (λ− µ)z[Ωαf(z)]′ + (1− λ+ µ)[Ωαf(z)],
D2,α
λ,µf(z) = Dα
λ,µ
(
D1,α
λ,µf(z)
)
,
Dn,α
λ,µf(z) = Dα
λ,µ
(
Dn−1,α
λ,µ f(z)
)
,
(1.11)
where λ ≥ µ ≥ 0, 0 ≤ α < 1 and n ∈ N.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
1220 E. DENIZ, H. ORHAN, J. SOKÓL
If f is given by (1.1), then from the definitions of the Dn,α
λ,µ and Ωα it is easy to see that
Dn,α
λ,µf(z) = z +
∞∑
k=2
Ψk,n(λ, µ, α)akz
k, (1.12)
where
Ψk,n(λ, µ, α) =
[
Γ(k + 1)Γ(2− α)
Γ(k + 1− α)
(1 + (λµk + λ− µ)(k − 1))
]n
. (1.13)
From (1.10) and (1.13), Dn,α
λ,µf(z) can be written, in terms of convolution as
Dn,α
λ,µf(z) = (ϕ(2, 2− α; z) ∗ gλ,µ(z))[n] ∗ f(z), (1.14)
where f ∗ . . . ∗ f︸ ︷︷ ︸
n times
= f [n] and
gλ,µ(z) =
z3(1− λ+ µ) + z2(λ− µ+ 2λµ− 2) + z
(1− z)3
= z +
∞∑
k=2
(1 + (λµk + λ− µ)(k − 1)) zk.
It should be remarked that the operator Dn,α
λ,µ is a generalization of many other linear operators
considered earlier. In particular, for f ∈ A we have the following:
(i) Dn,0
1,0 f(z) ≡ Dnf(z), the operator investigated by Salagean [32];
(ii) Dn,0
λ,0f(z) ≡ Dn
λf(z), the operator considered by Al-Oboudi [3];
(iii) D1,α
0,0 f(z) ≡ Ωαf(z), the fractional derivative operator studied by Owa and Srivastava [22];
(iv) Dn,0
λ,µf(z) ≡ Dn
λ,µf(z), the operator investigated by Răducanu and Orhan [28] (also see
Deniz and Orhan [7]);
(v) Dn,α
λ,0 f(z) ≡ Dn,α
λ f(z), the operator considered by Al-Oboudi and Al-Amoudi [4];
(vi) D1,α
λ,0f(z) ≡ Dα
λf(z), the operator studied by Noor, Arif and Ul-Haq [19].
Using the operator Dn,α
λ,µ , authors defined in [20] the classes β−UCVn,αλ,µ(γ) and β−SPn,αλ,µ(γ). For a
unified class of k-uniformly convex functions defined by the Dziok – Srivastava linear operator [23].
Definition 1.2. For λ ≥ µ ≥ 0, 0 ≤ α < 1, β ≥ 0, −1 ≤ γ < 1 and β + γ ≥ 0 a function
f ∈ A is said to be in the class β − UCVn,αλ,µ(γ) if it satisfies the following condition:
<
1 +
z
(
Dn,α
λ,µf(z)
)′′
(
Dn,α
λ,µf(z)
)′
> β
∣∣∣∣∣∣∣
z
(
Dn,α
λ,µf(z)
)′′
(
Dn,α
λ,µf(z)
)′
∣∣∣∣∣∣∣+ γ, z ∈ ∆. (1.15)
Definition 1.3. For λ ≥ µ ≥ 0, 0 ≤ α < 1, β ≥ 0, −1 ≤ γ < 1 and β + γ ≥ 0 a function
f ∈ A is said to be in the class β − SPn,αλ,µ(γ) if it satisfies the following condition:
<
z
(
Dn,α
λ,µf(z)
)′
Dn,α
λ,µf(z)
> β
∣∣∣∣∣∣∣
z
(
Dn,α
λ,µf(z)
)′
Dn,α
λ,µf(z)
− 1
∣∣∣∣∣∣∣+ γ, z ∈ ∆. (1.16)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
CLASSES OF ANALYTIC FUNCTIONS DEFINED BY A DIFFERENTIAL OPERATOR . . . 1221
Note that f ∈ β − SPn,αλ,µ(γ) if and only if Dn,α
λ,µf ∈ β − SP (γ). Using the Alexander type
relation, it is clear that
f ∈ β − UCVn,αλ,µ(γ)⇔ zf ′ ∈ β − SPn,αλ,µ(γ). (1.17)
Geometric interpretation. From (1.15) and (1.16), f ∈ β−UCVn,αλ,µ(γ) and g ∈ β−SPn,αλ,µ(γ) if
and only if p(z) = 1+zDn,α
λ,µf(z))′′/(Dn,α
λ,µf(z))′ and q(z) = z(Dn,α
λ,µg(z))′/Dn,α
λ,µg(z) take all its the
values in the domain Rβ,γ given in (1.4) which is included in the half plane <w > (β + γ)/(1 + β).
Thus we may rewrite the conditions (1.15) and (1.16) in the form
p ≺ P̂β,γ , q ≺ P̂β,γ , z ∈ ∆, (1.18)
where the function P̂β,γ given by (1.6).
By virtue of (1.15) and (1.16) and the properties of domain Rβ,γ , we have, respectively
<
1 +
z
(
Dn,α
λ,µf(z)
)′′
(
Dn,α
λ,µf(z)
)′
>
β + γ
1 + β
> 0, z ∈ ∆, (1.19)
and
<
z
(
Dn,α
λ,µf(z)
)′
Dn,α
λ,µf(z)
>
β + γ
1 + β
> 0, z ∈ ∆, (1.20)
which means that
f ∈ β − UCVn,αλ,µ(γ)⇒ Dn,α
λ,µf ∈ CV
(
β + γ
1 + β
)
⊆ CV (1.21)
and
f ∈ β − SPn,αλ,µ(γ)⇒ Dn,α
λ,µf ∈ ST
(
β + γ
1 + β
)
⊆ ST , (1.22)
where CV(γ), ST (γ), CV, ST denote the well-known classes of γ-convex, γ-starlike, convex and
starlike functions, respectively.
We note that by specializing the parameters n, α, λ, µ, β and γ, the class β −SPn,αλ,µ(γ) reduces
to several well-known subclasses of analytic functions. These subclasses are:
(a) 0−SP1,0
0,0(0) ≡ 0−SP0,α
λ,µ(0) ≡ ST and 0−UCV0,αλ,µ(0) ≡ 0−UCV1,00,0(0) ≡ 0−SP1,0
1,0(0) ≡
≡ CV (see [8, p. 40 – 43]),
(b) 0 − SP1,0
0,0(γ) ≡ 0 − SP0,α
λ,µ(γ) ≡ ST (γ) and 0 − UCV0,αλ,µ(γ) ≡ 0 − UCV1,00,0(γ) ≡ 0 −
− SP1,0
1,0(γ) ≡ CV(γ) (see [24]),
(c) 1− SP0,α
λ,µ(0) ≡ 1− SP1,0
0,0(0) ≡ SP (see [27]),
(d) 1− UCV0,αλ,µ(0) ≡ 1− UCV1,00,0(0) ≡ 1− SP1,0
1,0(0) ≡ UCV (see [9, 16]),
(e) β − SP0,α
λ,µ(0) ≡ β − SP1,0
0,0(0) ≡ β − SP (see [12]),
(f) β − UCV0,αλ,µ(0) ≡ β − UCV1,00,0(0) ≡ β − SP1,0
1,0(0) ≡ β − UCV (see [11]),
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
1222 E. DENIZ, H. ORHAN, J. SOKÓL
(g) 1−SP0,α
λ,µ(2ρ− 1) ≡ 1−SP1,0
0,0(2ρ− 1) ≡ PS∗(ρ) (0 ≤ ρ < 1) and 1−UCV0,αλ,µ(2ρ− 1) ≡
≡ 1− UCV1,00,0(2ρ− 1) ≡ 1− SP1,0
1,0(2ρ− 1) ≡ UCV(ρ) (see [2]),
(h) 1 − SP0,α
λ,µ(γ) ≡ 1 − SP1,0
0,0(γ) ≡ SP(γ) and 1 − UCV0,αλ,µ(γ) ≡ 1 − UCV1,00,0(γ) ≡ 1 −
− SP1,0
1,0(γ) ≡ UCV(γ) (see [26]),
(i) β − SP0,α
λ,µ(γ) ≡ β − SP1,0
0,0(γ) ≡ β − SP(γ) and β − UCV0,αλ,µ(γ) ≡ β − UCV1,00,0(γ) ≡
≡ β − SP1,0
1,0(γ) ≡ β − UCV(γ) (see [5]),
(j) 0− SPn,01,0 (γ) ≡ ST n(γ) (see [32]),
(k) β − SPn,01,0 (0) ≡ β − SPn (see [14, 17]),
(l) 0− SP1,α
0,0 (γ) ≡ ST α(γ) (see [33]),
(m) 1− SP1,α
0,0 (0) ≡ SPα (see [34]),
(n) β − SP1,α
0,0 (0) ≡ β − SPα (see [18]),
(o) β − SPn,αλ,0 (γ) ≡ SPnα,λ(β, γ) and β − UCVn,αλ,0 (γ) ≡ UCVnα,λ(β, γ) (see [4]).
For special values of parameters n, α, λ, µ, β and γ, from the general class β − SPn,αλ,µ(γ) and
the class β − UCVn,αλ,µ(γ), the following new classes can be obtained which are open questions:
β − SPn,0λ,µ(γ) ≡ β − SPnλ,µ(γ) and β − UCVn,0λ,µ(γ) ≡ β − UCVnλ,µ(γ),
0− SPn,αλ,µ(γ) ≡ ST n,αλ,µ(γ) and 0− UCV yn,αλ,µ (γ) ≡ CVn,αλ,µ(γ),
1− SPn,αλ,µ(0) ≡ SPn,αλ,µ and 1− UCVn,αλ,µ(0) ≡ UCVn,αλ,µ .
By (1.19) and (1.20), respectively, we note that β − UCVn,αλ,µ(γ) ⊆ CVn,αλ,µ
(
β + γ
1 + β
)
and β −
− SPn,αλ,µ(γ) ⊆ ST n,αλ,µ
(
β + γ
1 + β
)
.
In the present paper, basic properties of the classes β−UCVn,αλ,µ(γ) and β−SPn,αλ,µ(γ) are studied,
such as inclusion relations and coefficient bounds. Some interesting consequences of the main results
and their relevance to known results are also pointed out.
2. Inclusion relations. In this section, we are going to give several inclusion relationships for
the classes β−UCVn,αλ,µ(γ) and β−SPn,αλ,µ(γ), which are associated with the general linear multiplier
fractional differential operator Dn,α
λ,µ . To establish our main results, we shall require the following
lemmas.
Lemma 2.1 [30]. Let f and g be starlike of order 1/2. Then so is f ∗ g.
Lemma 2.2 [29, p. 54]. If f ∈ CV, g ∈ ST or f, g ∈ ST (1/2) , then for each function h
analytic in the unit disc ∆ we have
(f ∗ hg)(∆)
(f ∗ g)(∆)
⊆ coh(∆),
where coh(∆) denotes the closed convex hull of h(∆).
Lemma 2.3 [29, p. 60 – 61]. Suppose that 0 < b ≤ c. If c ≥ 2 or b+ c ≥ 3, then
ϕ(b, c; z) =
∞∑
k=0
(b)k
(c)k
zk+1, z ∈ ∆,
belongs to the class CV of convex functions.
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CLASSES OF ANALYTIC FUNCTIONS DEFINED BY A DIFFERENTIAL OPERATOR . . . 1223
Lemma 2.4. Let Ωαf be in the class β − SPn,αλ,µ(γ). Then f is in the class β − SPn,αλ,µ(γ).
Proof. Let Ωαf ∈ β − SPn,αλ,µ(γ). Then from (1.22), Dn,α
λ,µΩαf ∈ ST . Using (1.10) and (1.14),
we can write Dn,α
λ,µf in terms of Dn,α
λ,µΩαf as follows:
Dn,α
λ,µf(z) = ϕ(2− α, 2; z) ∗Dn,α
λ,µΩαf(z).
Moreover, ϕ(2 − α, 2; z) ∈ CV by Lemma 2.3 and so Dn,α
λ,µf is a starlike function as a convo-
lution of convex and starlike functions (see [29, p. 54]). So z/Dn,α
λ,µf(z) 6= 0 for z ∈ ∆ and
z(Dn,α
λ,µf(z))′/Dn,α
λ,µf(z) has no poles in ∆.
Furthermore, by convolution properties, we get
z(Dn,α
λ,µf(z))′ = ϕ(2− α, 2; z) ∗ z(Dn,α
λ,µΩαf(z))′.
Since ϕ(2− α, 2; z) ∈ CV and Dn,α
λ,µΩαf ∈ ST , using Lemma 2.2 we have
z
(
Dn,α
λ,µf(z)
)′
Dn,α
λ,µf(z)
=
ϕ(2− α, 2; z) ∗
[
z
(
Dn,α
λ,µΩαf(z)
)′
/
(
Dn,α
λ,µΩαf(z)
)]
Dn,α
λ,µΩαf(z)
ϕ(2− α, 2; z) ∗Dn,α
λ,µΩαf(z)
∈
∈ co
z
(
Dn,α
λ,µΩαf
)′(
Dn,α
λ,µΩαf
) (∆)
⊆ clRβ,γ .
Therefore, f ∈ β − SPn,αλ,µ(γ).
Lemma 2.4 is proved.
Corollary 2.1. Let Ωαf be in the class β − UCVn,αλ,µ(γ). Then f is in the class β − UCVn,αλ,µ(γ).
Proof. By virtue of (1.17) and Lemma 2.4, we obtain
Ωαf ∈ β − UCVn,αλ,µ(γ)⇔ z(Ωαf)′ ∈ β − SPn,αλ,µ(γ)⇔
⇔ Ωαzf ′ ∈ β − SPn,αλ,µ(γ)⇒ zf ′ ∈ β − SPn,αλ,µ(γ)⇔ f ∈ β − UCVn,αλ,µ(γ).
Therefore, f ∈ β − UCVn,αλ,µ(γ).
Corollary 2.1 is proved.
Lemma 2.5. Suppose that β + 2γ ≥ 1. If f ∈ β − SPn,αλ,µ(γ), then Dn,α
λ,µf ∈ ST (1/2) .
Proof. The result follows immediately from (1.20) whenever (β + γ)/(1 + β) ≥ 1/2.
Theorem 2.1. If[
0 < λ ≤ 1 +
√
5
2
and 0 < µ and λ− 1 ≤ µ ≤ λ
1 + λ
]
or [0 = λ = µ] or [0 = µ < λ],
then
β − SPn+1,α
λ,µ (γ) ⊆ β − SPn,αλ,µ(γ). (2.1)
Proof. In the proof we will use several convolution results, see for example [29]. In this proof,
for simplicity let us denote ϕ(2, 2−α; z) = ϕ, gλ,µ = g. From (1.18) and (1.14) it is easy to see that
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1224 E. DENIZ, H. ORHAN, J. SOKÓL
f ∈ β − SPn,αλ,µ ⇔
z
[
Dn,α
λ,µf(z)
]′
Dn,α
λ,µf(z)
≺ P̂β,γ(z)⇔
z
[
(ϕ ∗ g)[−1] ∗Dn+1,α
λ,µ f(z)
]′
(ϕ ∗ g)[−1] ∗Dn+1,α
λ,µ f(z)
≺ P̂β,γ(z)⇔
⇔
(ϕ ∗ g)[−1] ∗ z
[
Dn+1,α
λ,µ f(z)
]′
(ϕ ∗ g)[−1] ∗Dn+1,α
λ,µ f(z)
≺ P̂β,γ(z), (2.2)
where the convex function P̂β,γ(z) = 1 + P1z + . . . is given by (1.6) and (ϕ ∗ g)[−1] = ϕ[−1] ∗ g[−1]
denotes the convolution inverse with respect to ϕ ∗ g. Recall that f [−1] is the convolution inverse to
f if f ∗ f [−1] = z/(1− z). Now we will to show that g[−1] is convex. If λµ > 0, then we have
(ϕ ∗ g)[−1](z) = ϕ[−1](z) ∗ g[−1](z) = ϕ[−1](z) ∗
[ ∞∑
k=1
1
1 + (k − 1)(λµk + λ− µ)
zk
]
=
= ϕ[−1](z) ∗
[
1
(1 + k1)
√
µλ
∞∑
k=1
1 + k1
k + k1
zk
]
∗
[
1
(1 + k2)
√
µλ
∞∑
k=1
1 + k2
k + k2
zk
]
=
= ϕ[−1](z) ∗ g[−1]1 (z) ∗ g[−1]2 (z), (2.3)
where
ki =
λ− µ− λµ±
√
(λ− µ− λµ)2 − 4λµ(1 + µ− λ)
2λµ
, i = 1, 2.
Observe that k1, k2 have a positive real under assumptions of Theorem 2.1. For <(x) ≥ 0 or x = 0
the function
h̃(x; z) =
∞∑
k=1
(1 + x)
(k + x)
zk, z ∈ ∆, (2.4)
is convex univalent [30]. So g
[−1]
1 and g
[−1]
2 in (2.3) are convex when λµ > 0. Otherwise, if
µ = λ = 0 or if 0 = µ < λ, then it is easy to see that g[−1] has the form of the type (2.4) so it is
convex too. In the famous paper [31] it was proved the Polya – Schoenberg conjecture that the class
of convex univalent functions is preserved under convolution. Under our assumptions on α, λ, µ the
function g[−1] = g
[−1]
1 ∗g[−1]2 is convex as the convolution of two convex functions and by Lemma 2.3
ϕ[−1](z) = (ϕ(2, 2− α; z))[−1] = z +
∞∑
k=1
(2− α)k
(2)k
zk
is a convex function too. Therefore (ϕ ∗ g)[−1] is the convex function. Let f ∈ β −SPn+1,α
λ,µ (γ). By
Definition 1.3 we have
z(Dn+1,α
λ,µ f(z))′
Dn+1,α
λ,µ f(z)
= P̂β,γ(ω(z)), (2.5)
where ω is an analytic function with ω(0) = 0 and |ω(z)| < 1 for z ∈ ∆. From (2.5) we have that
Dn+1,α
λ,µ f is a starlike function. Therefore, by (2.3) and by (2.5) we obtain from (2.2)
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CLASSES OF ANALYTIC FUNCTIONS DEFINED BY A DIFFERENTIAL OPERATOR . . . 1225
f ∈ β − SPn,αλ,µ(γ)⇔
(ϕ ∗ g)[−1] ∗
(
Dn+1,α
λ,µ f(z)
)
P̂β,γ(ω(z))
(ϕ ∗ g)[−1] ∗Dn+1,α
λ,µ f(z)
≺ P̂β,γ(z). (2.6)
The function P̂β,γ is univalent so the subordination principle and Lemma 2.2 show that the last
subordination is true and so f ∈ β − SPn,αλ,µ(γ).
Theorem 2.1 is proved.
Corollary 2.2. Let α ∈ [0, 1) and n ∈ N. Under the conditions stated in Theorem 2.1 we have
β − SPn,αλ,µ(γ) ⊆ β − SP(γ).
Proof. Suppose that f ∈ β − SPn,αλ,µ(γ). Then as in the proof of Theorem 2.1 we obtain
zf ′(z)
f(z)
=
[
(ϕ ∗ g)[−1](z)
][n] ∗ (Dn,α
λ,µf(z)
)
P̂β,γ(ω(z))[
(ϕ ∗ g)[−1](z)
][n] ∗Dn,α
λ,µf(z)
≺ P̂β,γ(z).
Therefore, f ∈ β − SP(γ).
Corollary 2.2 is proved.
By (1.17) and Theorem 2.1, we deduce the next consequences.
Corollary 2.3. Let α ∈ [0, 1) and n ∈ N. Under the assumptions in Theorem 2.1 we have
β − UCVn+1,α
λ,µ (γ) ⊆ β − UCVn,αλ,µ(γ).
Proof. From (1.17) and Theorem 2.1, we get
f ∈ β − UCVn+1,α
λ,µ (γ)⇔ zf ′ ∈ β − SPn+1,α
λ,µ (γ)⇔ zf ′ ∈ β − SPn,αλ,µ(γ)⇔ f ∈ β − UCVn,αλ,µ(γ).
Thus f ∈ β − UCVn,αλ,µ(γ).
Corollary 2.3 is proved.
Corollary 2.4. Let α ∈ [0, 1) and n ∈ N. Under the conditions stated in Theorem 2.1 we have
β − UCVn,αλ,µ(γ) ⊆ β − UCV(γ).
Remark 2.1. (1) Taking γ = α = µ = 0 and λ = 1 in Theorem 2.1, we get the result by Kanas
and Yaguchi [13].
(2) Taking µ = 0 in Theorem 2.1, we get a result due to Al-Oboudi and Al-Amoudi [4].
Remark 2.2. For special values of parameters n, α, β and γ and for λ, µ satisfying the assump-
tions of Theorem 2.1 we obtain the following new results:
(1) β − SPn+1
λ,µ (γ) ⊆ β − SPnλ,µ(γ) and β − UCVn+1
λ,µ (γ) ⊆ β − UCVnλ,µ(γ),
(2) ST n+1,α
λ,µ (γ) ⊆ ST n,αλ,µ(γ) and CVn+1,α
λ,µ (γ) ⊆ CVn,αλ,µ(γ),
(3) SPn+1,α
λ,µ ⊆ SPn,αλ,µ and UCVn+1,α
λ,µ ⊆ UCVn,αλ,µ .
Theorem 2.2. Suppose that 0 ≤ δ ≤ α < 1. Then
β − SPn,αλ,µ(γ) ⊆ β − SPn,δλ,µ(γ),
whenever β + 2γ ≥ 1.
Proof. Let f ∈ β − SPn,αλ,µ(γ). Then by (1.14) and convolution properties, we get
Dn,δ
λ,µf(z) = [ϕ(2, 2− δ; z) ∗ gλ,µ(z)][n] ∗ f(z) = [ϕ(2− α, 2− δ; z)][n] ∗Dn,α
λ,µf(z)
and
z(Dn,δ
λ,µf(z))′ = [ϕ(2− α, 2− δ; z)][n] ∗ z(Dn,α
λ,µf(z))′.
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1226 E. DENIZ, H. ORHAN, J. SOKÓL
Also, it is known [15] that ϕ(2 − α, 2 − δ; z) ∈ ST (1/2) . So by applying Lemma 2.1 we have
[ϕ(2− α, 2− δ; z)][n] ∈ ST (1/2) . For β + 2γ ≥ 1, we have by Lemma 2.5 Dn,α
λ,µf ∈ ST (1/2) .
Using Lemma 2.2, we obtain
z
(
Dn,δ
λ,µf(z)
)′
Dn,δ
λ,µf(z)
=
[ϕ(2− α, 2− δ; z)][n] ∗
[
z
(
Dn,α
λ,µf(z)
)′
/
(
Dn,α
λ,µf(z)
)]
Dn,α
λ,µf(z)
[ϕ(2− α, 2− δ; z)][n] ∗Dn,α
λ,µf(z)
∈
∈ co
z
(
Dn,α
λ,µf
)′(
Dn,α
λ,µf
) (∆)
⊆ clRβ,γ .
Therefore, f ∈ β − SPn,δλ,µ(γ).
Theorem 2.2 is proved.
Corollary 2.5. Let β + 2γ ≥ 1. Then β − SPn,αλ,µ(γ) ⊆ β − SPnλ,µ(γ).
The proof of the following Corollary 2.6 runs parallel to that of Corollary 2.3, and we omit the
details.
Corollary 2.6. Let 0≤ δ ≤ α < 1. Then β − UCVn,αλ,µ(γ) ⊆ β − UCVn,δλ,µ(γ) ⊆ β − UCVn,0λ,µ(γ),
where β + 2γ ≥ 1.
Remark 2.3. (1) Taking β = λ = µ = 0 and n = 1 in Theorem 2.2, we get the result of
Srivastava, Mishra and Das [33].
(2) Taking γ = λ = µ = 0 and β = n = 1 in Theorem 2.1, we get the result of Srivastava and
Mishra [34].
(3) Taking γ = λ = µ = 0 and n = 1 in Theorem 2.1 and Corollary 2.6, we get the result by
Mishra and Gochhayat [18].
(4) Taking µ = 0 in Theorem 2.1 and Corollary 2.6, we get the result of Al-Oboudi and Al-
Amoudi [4].
Remark 2.4. For special values of parameters n, α, β and γ, we arrive the following new results
for 0 ≤ δ ≤ α < 1:
(1) β − SP1,α
0,0 (γ) ⊆ β − SP1,δ
0,0(γ) and β − UCV1,α0,0 (γ) ⊆ β − UCV 1,δ
0,0(γ) for β + 2γ ≥ 1,
(2) ST n,αλ,µ(γ) ⊆ ST n,δλ,µ(γ) and CVn,αλ,µ(γ) ⊆ CVn,δλ,µ(γ) for 1/2 ≤ γ < 1,
(3) SPn,αλ,µ ⊆ SP
n,δ
λ,µ and UCVn,αλ,µ ⊆ UCV
n,δ
λ,µ.
From (1.15) and (1.16) we directly obtain the following useful Theorem 2.3.
Theorem 2.3. If β1 ≥ β2, γ1 ≥ γ2 then β1 − SPn,αλ,µ(γ1) ⊆ β2 − SPn,αλ,µ(γ2) and β1 −
− UCVn,αλ,µ(γ1) ⊆ β2 − UCVn,αλ,µ(γ2).
Remark 2.5. (1) By putting α = µ = 0, λ = 1 and γ1 = γ2 = 0 in Theorem 2.3 for the
class β − SPn,αλ,µ(γ), we obtain β1 − SPn ⊆ β2 − SPn, which was asserted earlier by Kanas and
Yaguchi [13].
(2) Taking µ = 0 in Theorem 2.3, we get the result of Al-Oboudi and Al-Amoudi [4].
Corollary 2.7. Under the conditions stated in Theorem 2.1 we have β − SPn,αλ,µ(γ) ⊆ β −
− SP1,α
0,0 (γ) ⊆ SPα for β ≥ 1.
Proof. Let f be in β −SPn,αλ,µ(γ). Then f belongs to β −SP1,α
λ,µ(γ) by applying Theorem 2.1.
By the same steps of the proof of Theorem 2.1 we have Ωαf ∈ β−SP(γ) and by using Theorem 2.3,
Ωαf ∈ SP for β ≥ 1. Thus f ∈ SPα.
Corollary 2.8. Under the conditions stated in Theorem 2.1 we have β − UCVn,αλ,µ(γ) ⊆ β −
− UCV1,α0,0 (γ) ⊆ 1− UCV1,α0,0 (0) for β ≥ 1.
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CLASSES OF ANALYTIC FUNCTIONS DEFINED BY A DIFFERENTIAL OPERATOR . . . 1227
Theorem 2.4. Let f ∈ β − SPn,αλ,µ(γ) and h ∈ CV. Then
f(z) ∗ h(z) ∈ β − SPn,αλ,µ(γ).
Proof. Let f ∈ β − SPn,αλ,µ(γ). To prove the required result, it is sufficient to prove that
z(h(z) ∗Dn,α
λ,µf(z))′
h(z) ∗Dn,α
λ,µf(z)
∈ Rβ,γ , z ∈ ∆.
Theorem 2.4 is proved.
The remaining part of the proof of Theorem 2.4 is similar to that of Lemma 2.4 and hence we
omit it.
Remark 2.6. Taking α = µ = λ = 0, γ = 2ρ − 1, 0 ≤ ρ < 1, n = 1 in Theorem 2.4, we get
the result by Ali [2].
3. Coefficient bounds. In the following we give the bounds for the coefficients of series
expansion of functions belonging to the classes β −SPn,αλ,µ(γ) and β −UCVn,αλ,µ(γ) and sufficient for
a function to be in these classes.
Let P̂β,γ be given in (1.6) and let fβ,γ be defined by
P̂β,γ(z) =
z(Dn,α
λ,µfβ,γ(z))′
Dn,α
λ,µfβ,γ(z)
, z ∈ ∆. (3.1)
The function fβ,γ is in the class β − SPn,αλ,µ(γ) and if we denote
P̂β,γ(z) = 1 + P1z + . . . , fβ,γ(z) = z +A2z
2 + . . . ,
then in view of (1.12) and (3.1), we have a coefficient relation
(k − 1)AkΨk,n(λ, µ, α) =
k−1∑
j=1
Pk−jAjΨj,n(λ, µ, α), A1 = 1, Ψ1,n(λ, µ, α) = 1. (3.2)
In particular, by a straightforward computation we obtain
A2 =
1
Ψ2,n(λ, µ, α)
P1, (3.3)
observe also, that the coefficients Ak are nonnegative because Ψk,n(λ, µ, α) ≥ 0 and Pk are nonne-
gative (for Taylor series expansion of P̂β,γ , see [14, 16, 26]).
As simple consequence of the above and the result given in [12], we give sharp bound on the
second coefficient for functions of the class β − SPn,αλ,µ(γ).
Theorem 3.1. If a function f of the form (1.1) is in β − SPn,αλ,µ(γ), then
|ak| ≤
1
Ψk,n(λ, µ, α)
(P1)k−1
(k − 1)!
, k ≥ 2, (3.4)
where
P1 := P1(β, γ) =
8(1− γ)(arccosβ)2
π2(1− β2)
, 0 ≤ β < 1,
8(1− γ)
π2
, β = 1,
π2(1− γ)
4
√
t(1 + t)(β2 − 1)K2(t)
, β > 1.
(3.5)
The result is sharp for k = 2 or β = 0.
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1228 E. DENIZ, H. ORHAN, J. SOKÓL
For the proof of this theorem, we need the following result by Rogosinski [25].
Rogosinski’s theorem [25]. Let h(z) = 1 +
∑∞
k=1
ckz
k be subordinate to H(z) = 1 +
+
∑∞
k=1
Ckz
k in ∆. If H(z) is univalent in ∆ and H(∆) is convex, then |ck| ≤ |C1| for k ≥ 1.
Proof of Theorem 3.1. Let f ∈ β − SPn,αλ,µ(γ), f(z) = z +
∑∞
k=2
akz
k. By (1.18), we obtain
z(Dn,α
λ,µf(z))′
Dn,α
λ,µf(z)
≺ P̂β,γ(z), z ∈ ∆.
Define h(z) =
z(Dn,α
λ,µf(z))′
Dn,α
λ,µf(z)
= 1 +
∑∞
k=1
ckz
k. The function P̂β,γ is univalent in ∆ and P̂β,γ(∆)
is the is convex conic domain so Rogosinski’s theorem applies. Then we have
|ck| ≤ |P1| = P1, k ≥ 1, (3.6)
where P1 = P1(β, γ) is given by (3.5). Now writing z(Dn,α
λ,µf(z))′ = h(z)Dn,α
λ,µf(z) and comparing
coefficients of zk on both sides, we get
(k − 1)akΨk,n(λ, µ, α) =
k−1∑
j=1
ck−jajΨj,n(λ, µ, α), a1 = 1, Ψ1,n(λ, µ, α) = 1. (3.7)
From (3.6) and (3.7) we get |a2| =
1
Ψ2,n(λ, µ, α)
|c1| ≤
P1
Ψ2,n(λ, µ, α)
. So the result is true for k = 2.
Let k ≥ 2 and assume that the inequality (3.4) is true for all j ≤ k − 1. By using (3.6), (3.7) and
applying the induction hypothesis to |aj | , we get
|ak| ≤
1
(k − 1)Ψk,n(λ, µ, α)
|c1|+ k−1∑
j=2
|ck−j | |aj |Ψj,n(λ, µ, α)
≤
≤ P1
(k − 1)Ψk,n(λ, µ, α)
1 +
k−1∑
j=2
(P1)j−1
(j − 1)!
.
By applying mathematical induction another time, we find that
1 +
k−1∑
j=2
(P1)j−1
(j − 1)!
=
(1 + P1)(2 + P1) . . . ((k − 2) + P1)
(k − 2)!
. (3.8)
Thus we get the inequality (3.4). In view of (3.3) the result is sharp for k = 2. If β = 0 then
Pk(0, γ) = P1(0, γ) = 2(1− γ), k = 1, 2, . . . , and in view of (3.2) we have
Ak =
1
Ψk,n(λ, µ, α)
(P1)k−1
(k − 1)!
, k ≥ 2.
Applying the relation (1.17), we observe that the extremal function of β − UCVn,αλ,µ(γ) denoted by
Fβ,γ(z), is given by
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CLASSES OF ANALYTIC FUNCTIONS DEFINED BY A DIFFERENTIAL OPERATOR . . . 1229
Fβ,γ(z) = Fβ,γ(z) = z +B2z
2 + . . . =
z∫
0
fβ,γ(ξ)
ξ
dξ,
where fβ,γ(z) is defined by (3.1). By (3.3) we get
B2 =
1
2Ψ2,n(λ, µ, α)
P1.
Theorem 3.1 is proved.
Repeating similar consideration as given in the proof of Theorem 3.1 and applying relation (1.17)
we can prove the next two results.
Corollary 3.1. If a function f of the form (1.1) is in β − UCVn,αλ,µ(γ), then
|ak| ≤
1
Ψk,n(λ, µ, α)
(P1)k−1
(k)!
, k ≥ 2,
where P1 := P1(β, γ) is given by (3.5). The result is sharp for k = 2 or β = 0.
Corollary 3.2.
⋂∞
n=1
β − SPn,αλ,µ(γ) = {z}.
Proof. Suppose that there exists f(z) = z +
∑∞
k=2
akz
k belonging to β − SPn,αλ,µ(γ) for all
n ∈ N. Then Theorem 3.1 gives |ak| ≤
(2− α)n
(2)n
(P1)k−1
(k − 1)!
for all n ∈ N. The sequence on the
right-hand side is decreasing to 0 with respect to n. Thus ak = 0, for all k ≥ 2.
Remark 3.1. (1) Putting α = µ = λ = β = 0, n = 1 in Theorem 3.1 and Corollary 3.1 we get
the results which, in turn, yields the corresponding results given earlier by Robertson [24].
(2) For special values of the parameters (α = µ = β = 0 andλ = 1) or (λ = µ = β = 0
and n = 1) in Theorem 3.1, we obtain the results of Sălăgean [32] or Srivastava and Mishra [33],
respectively, which are sharp results.
(3) Taking α = µ = λ = γ = 0, β = n = 1 in Theorem 3.1, we get the result by Rønning [27].
(4) Setting α = µ = λ = γ = 0, n = 1 in Theorem 3.1, we get the result by Kanas and
Wiśniowska [12].
(5) Taking α = µ = γ = 0, λ = 1 in Theorem 3.1, we get the result by Kanas and Yaguchi [13].
(6) Upon setting α = µ = λ = γ = 0, n = 1 in Corollary 3.1, we obtain the result which is an
improvement of a result due to Kanas and Wiśniowska [10].
(7) For µ = 0 Theorem 3.1 and Corollary 3.1 would lead us, respectively, to the corresponding
results obtained by Al-Oboudi and Al-Amoudi [4].
Remark 3.2. For special values of the parameters n, α, µ, λ and β in Theorem 3.1 and Corollary
3.1, we get the coefficients bounds which is a new result for the classes β−SPnλ,µ(γ), β−UCVnλ,µ(γ),
ST n,αλ,µ(γ), CVn,αλ,µ(γ),β − SP1,α
0,0 (γ) and β − UCV1,α0,0 (γ).
Now for functions in the class β − SPn,αλ,µ(γ), we establish the following result.
Theorem 3.2. A function f of the form (1.1) is in β − SPn,αλ,µ(γ) whenever
∞∑
k=2
[k(1 + β)− (β + γ)] |ak|Ψk,n(λ, µ, α) ≤ 1− γ. (3.9)
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1230 E. DENIZ, H. ORHAN, J. SOKÓL
Proof. From (3.9) we can find that 1−
∑∞
k=2
Ψk,n(λ, µ, α) |ak| > 0. Thus we get
β
∣∣∣∣∣∣∣
z
(
Dn,α
λ,µf(z)
)′
Dn,α
λ,µf(z)
− 1
∣∣∣∣∣∣∣−<
z
(
Dn,α
λ,µf(z)
)′
Dn,α
λ,µf(z)
− 1
≤ (1 + β)
∣∣∣∣∣∣∣
z
(
Dn,α
λ,µf(z)
)′
Dn,α
λ,µf(z)
− 1
∣∣∣∣∣∣∣ ≤
≤
(1 + β)
∑∞
k=2
(k − 1)Ψk,n(λ, µ, α) |ak| |z|k−1
1−
∑∞
k=2
Ψk,n(λ, µ, α) |ak| |z|k−1
<
<
(1 + β)
∑∞
k=2
(k − 1)Ψk,n(λ, µ, α) |ak|
1−
∑∞
k=2
Ψk,n(λ, µ, α) |ak|
.
This last expression is bounded above by (1− γ) if (3.9) is satisfied. Therefore f ∈β − SPn,αλ,µ(γ).
By virtue of (1.17) and Theorem 3.2, we have the following corollary.
Corollary 3.3. A function f of the form (1.1) is in β − UCVn,αλ,µ(γ) whenever
∞∑
k=2
[k(1 + β)− (β + γ)]k |ak|Ψk,n(λ, µ, α) ≤ 1− γ.
Corollary 3.4. If |a2| ≤
1− γ
(2 + β − γ)Ψk,n(λ, µ, α)
, then f(z) = z + a2z
2 belongs to the class
β − SPn,αλ,µ(γ).
Remark 3.3. (1) If we consider α = µ = γ = 0, β = 1, λ = 1 in Theorem 3.2 and Corollary 3.3,
we obtain the same result by Bharti, Parvatham and Swaminathan [5].
(2) Taking α = µ = γ = 0, λ = 1 in Theorem 3.2, we get the result by Kanas and Yaguchi [13].
(3) Taking α = µ = λ = 0, γ = 2ρ− 1, 0 ≤ ρ < 1, n = 1 in Theorem 3.2, we get the result by
Ali [2].
For µ = 0, Theorem 3.2 and Corollary 3.3 would lead us, respectively, to the corresponding
results obtained by Al-Oboudi and Al-Amoudi [4].
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Received 31.12.12,
after revision — 14.05.14
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