Volume 2010, Article ID 275935,12pages doi:10.1155/2010/275935
Research Article
On Certain Classes of
p
-Valent Functions by Using
Complex-Order and Differential Subordination
Abdolreza Tehranchi
1and Adem Kılıc¸man
21Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran
2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM),
Serdang, Selangor 43400, Malaysia
Correspondence should be addressed to Adem Kılıc¸man,[email protected]
Received 29 May 2010; Revised 24 September 2010; Accepted 16 October 2010
Academic Editor: Vladimir Mityushev
Copyrightq2010 A. Tehranchi and A. Kılıc¸man. 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.
The aim of the present paper is to study thep-valent analytic functions in the unit disk and satisfy the differential subordinationszIpr, λfzj1/p−jIpr, λfzj≺a aB A−
Bβz/a1Bz,whereIpr, λis an operator defined by S˘al˘agean andβis a complex number. Further we define a new related integral operator and also study the Fekete-Szego problem by proving some interesting properties.
1. Introduction
LetAbe the class of analytic functions inΔ {z∈C : |z|<1}. LetApdenote the class of all analytic functions in the form of
fz ezp−
2p−1
np−1
tn−p1zn−p1 2F1a, b;c;z, |z|<1, 1.1
whereF1a, b;c;zis Gaussian hypergeometric function defined by
2F1a, b;c;z ∞
n0
a, nb, n
c, nn! z n,
a, n Γan
Γa aa1, n−1, c > b >0, c > ab,
tn−p1
a, n−p1b, n−p1
c, n−p1n−p1! , e >0.
Note that it is easy to see that these functions are analytic in the unit diskΔ; for more details on hypergeometric functions2F1a, b;c.z, see 1,2.
Definition 1.1. A functionf ∈ Apis said to be in the classS∗pα,p-valently starlike functions of orderα, if it satisfies Re{zfz/fz}> α,0≤α < p, z∈Δ. We writeS∗p0 S∗p, the class ofp-valently starlike functions inΔ.
Similarly, a functionf∈ Apis said to be in the classCpα,p-valently convex of order α, if it satisfies Re{1zfz/fz}> α, 0≤α < p, z∈Δ.
Lethzbe analytic andh0 p. A functionf∈ Apis in the classS∗phif
zfz
fz ≺hz, z∈Δ. 1.3
The classS∗phand a corresponding convex classCphwere defined by Ma and Minda in 3. Similar results which are related to the convex class can also be obtained easily from the corresponding functions inS∗ph. For example,
iifp1 and
hz 1z
1−z, 1.4
then the classes reduce to the usual classes of starlike and convex functions;
iiifhz 1 1−2αz/1−zwhere 0≤α <1, then the classes are reduced to the usual classes of starlike and convex functions of orderα;
iiiifhz p1Az/1Bz, where−1≤B < A≤1, then the classes are reduced to the class of Janowski starlike functionsS∗p A, Bwhich is defined by
S∗p A, B
f∈ Ap: zf
f ≺p 1Az
1Bz, −1≤B < A≤1, z∈Δ
; 1.5
ivif hz 1z/1−zα where p 1 and 0 < α ≤ 1, then the classes reduce to the classes of strongly starlike and convex functions of orderαthat consists of univalent functionsf ∈ Asatisfing
arg
zfz
fz < απ
2 , 0< α≤1, z∈Δ 1.6
or equivalently we have
SS∗α
f ∈ Ap: zf
f ≺
1z 1−z
α
, 0< α≤1, z∈Δ
In the literature, there are several works and many researchers have been studying the related problems. For example, Obradovi´c and Owa 4, Silverman 5, Obrad-owiˇc and Tuneski 6, and Tuneski 7have studied the properties of classes of func-tions which are defined in terms of the ratio of 1zfz/fzandzfz/fz.
Definition 1.2. A functionf∈ Apis said to bep-valent Bazilevic of typeηand orderαif there exists a functiong∈S∗psuch that
Re
zfz f1−ηzgηz
> α z∈Δ 1.8
for someη η ≥ 0andα0 ≤ α < p. We denote byBpη, α, the subclass ofApconsisting of all such functions. In particular, a function in Bp1, α Bpαis said to be p-valently close-to-convex of orderαinΔ.
Definition 1.3. Letfandg be analytic functions inΔ, then we sayf is subordinate togand denoted byf ≺ gif there exists a Schwarz functionwz, analytic inΔwithw0 0 and
|wz|<1, such thatfz gwz, z∈Δ. In particular, if the functiongis univalent inΔ, the above subordination is equivalent tof0 g0andfΔ⊂gΔ. Also, we say thatgis superordinate tof; see 8.
Definition 1.4. Motivated by the multiplier transformation on A, we define the operator
Ipr, λ; by the following infinite series whenfz zp∞np1anznthen
Ipr, λfz zp ∞
n1p
nλ pλ
r
anzn λ≥0. 1.9
S˘al˘agean derivative operator is closely related to the operator Ipr, λ; see 9. In 10, Uralegaddi and Somanatha also studied the caseI1r,1 Ir. The operatorI1r, λ Iλr was studied recently by Cho and Srivastava 11and Cho and Kim 12.
Definition 1.5. Differential operator, for eachfz zp∞
np1anzn, we have
fjz p! p−j!z
p−j ∞ n1p
n!
n−j!anz
n−j, 1.10
wheren, p∈N, p > j, andj∈N0{0} ∪N. In particular, ifj0 we havef0z fz.
Definition 1.6. A function f ∈ Ap is said to be in the class Apλ, r, j;h if it satisfies the following subordination:
zIpr, λfz
j1
p−jIpr, λfz
and in this study we consider
hz 1A−B a
βz
1Bz, z∈Δ, 1.12
where−1 ≤B < A ≤1, a > 0 andβ/0is a complex number; so we denoteApλ, r, j;h
Apλ, r, j, β, a, A, B. Then we say that fz is superordinate to hz if fz satisfies the following:
hz≺ z
Ipr, λfz
j1
p−jIpr, λfz
j, 1.13
wherehzis analytic inΔandh0 1. Further we note that if
zIpr, λfz
j1
Ipr, λfz
j ≺
p−jaaB A−Bβz
a1Bz hz. 1.14
By choosingj r 0, p 1, soh0 1, thenfz ∈ S∗h. Fora A β 1, B −1, andp ≥ 1, we havefz ∈S∗p1. But ifa β 1 andj r 0 and−1 ≤ B < A ≤ 1, then fz ∈ S∗ A, B, a class of Janowski starlike functions. If we putp a β A 1, B
−1, thenfz ∈SS∗1classes of strongly starlike. ByDefinition 1.2, ifgz ∈S∗, univalent starlike, andj r0 andpaAβ1,B−1 and ifw Re{zfz/fzg2z}>1, then
fz∈ B2,1is a class Bazilevic functions of typeη2 and orderα1.
2. Main Results
Theorem 2.1. Let the functionfzbe of the form1.1. If someA, B,−1< B < A≤1, andβ/0
are complex numbers and
∞
mp1
γλrm, pa1Bδm, j1−δm, jp−j−δm, jA−Bp−jβkm
<βeA−Bδp, j1,
2.1
thenfz∈ Apλ, r, j, β, a, A, B, whereγλrm, p mλ/pλr, δm, j m!/m−j! and forr, j∈N0, λ >0, p∈N, j < p. The result is sharp.
Proof. Since the functionfzin the theorem can be expressed in the form
fz ezp ∞
n2p
kn−p1zn−p1 or fz ezp ∞
mp1
wheremn−p1 andkm a, mb, m/c, mm!, and also we have for allr, j∈ N0,
Ipr, λfz
j ep! p−j!z
p−j ∞ mp1
mλ pλ
r m!
m−j!kmz m−j
eδp, jzp−j ∞
mp1
γλrm, pδm, jkmzm−j,
2.3
now, assume that the condition 2.1 holds true. We show that f ∈ Apλ, r, j, β, a, A, B. Equivalently, we prove that
az
Ipr, λfz
j1−ap−jI
pr, λfz
j
p−jRIpr, λfz
j−BazI
pr, λfz
j1
<1, 2.4
whereRaB A−Bβ. But we have
az
Ipr, λfz
j1−
ap−jIpr, λfz
j
p−jRIpr, λfz
j−
BazIpr, λfz
j1
a∞mp1γr λ
m, pδm, j1−δm, jp−jkmzm−j
βA−Bδp, j1ezp−j−∞mp1γλrm, paBC −δm, jA−Bβp−jkmzm−j
< ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
a∞mp1γλrm, pδm, j1−δm, jp−jkm
βeA−Bδ
p, j1−∞mp1γλrm, paBC −δm, jA−Bβp−jkm
⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ <1. 2.5
whereCdenotesδm, j1−δm, jp−j.
The last inequality is true by2.1and this completes the proof. The result is sharp for the functionsfmzdefined inΔby
fmz
ezp βeA−Bδ
p, j1
γλrm, pa1Bδm, j1−δm, jp−j−δm, jA−Bp−jβz m
2.6
Remark 2.2. We observe that ifB /0, the converse of the above theorem needs not be true. For instance, consider the functionfzdefined by
zIpr, λfz
j1
p−jIpr, λfz
j ≺
a−SgnBpB A−Bβz
a1−SgnBBz , 2.7
where SgnB 1,0,−1 thus accordinglyB > 0, B 0 and B < 0. It is easily seen that fz∈ Apλ, r, j, β, a, A, Band
km−
1BβeA−Bδp, j1SgnBm−p−1Bm−p−2
a1Bδm, j1δm, jp−jδm, jA−Bp−jβγλrm, p
m≥p1
2.8
so that
∞
mp1
γλrm, p
a1Bδm, j1−δm, jp−j−βA−Bδm, jp−j βeA−Bδp, j1
|km|
1B ∞
mp1
Bm−p−2 1B
1−B >1,
2.9
whereA,Bare satisfying the conditions−1≤B < A≤1, 0< B <1. This establishes our claim.
Theorem 2.3. If the functionfz∈ Apλ, r, j, β, a, A, B, then
|km| ≤
βA−Bδp, j1e aγr
λ
m, pδm, j1−δm, jp−j, m≥p−1, 2.10
Proof. We have
zIpr, λfz
j1
p−jIpr, λfz
j
aaB A−Bβwz
a1Bwz , 2.11
wherewz ∞ ip1wi−pz
i−pis defined as in theDefinition 1.3. Now we can write
a ∞
ip1
γr λ
i, pδi, j1−δi, jp−jkizi−j
⎧ ⎨
⎩βA−Bδ
p, j1ezp−j− ∞
ip1
γλri, pδi, j1Ba−δi, jp−jRkizi−j
⎫ ⎬ ⎭
∞
ip1
wi−pzi−p,
2.12
whereRaB A−Bβ. Now if we equalize the coefficients of the same power ofzin both sides, then we have
a m
ip−1
γλri, pδi, j1−δi, jp−jkizi−j ∞
im1
cizi−j
⎧ ⎨
⎩βA−Bδ
p, j1ezp−j− m−1
ip1
γλri, pδi, j1Ba−δi, jp−jRkizi−j
⎫ ⎬ ⎭wz,
2.13
where ci’s are suitable constants. By multiplying each side of the above equation by its conjugate and letting|z|1,r → 1−, we get
a2 m
ip−1
γλ2ri, pδi, j1−δi, jp−j2|ki|2
≤βA−Bδp, j1e2 m−1
ip1
γλ2ri, pδi, j1Ba−δi, jp−jR2|ki|2
2.14
so that
a2γλ2rm, pδm, j1−δm, jp−j2|km|2
≤βA−Bδp, j1e2−1−a2 m−1
ip1
γλ2ri, pδi, j1Ba−δi, jp−jR2|ki|2.
Since−1< B < A≤1 and 0< a < A−B <1, we have
|km| ≤
βA−Bδp, j1e aγr
λ
m, pδm, j1−δm, jp−j, m≥p−1 2.16
and this completes the proof. Note that the estimate in2.10is sharp for the functionsfmz defined inΔ; whenjr 0 in1.3, then
fmz exp
z
0
pψtRa t1Bψtdt
, m≥1p, 2.17
where|ψt|<1,z∈ΔandRaB A−Bβ. We can chooseψt tm.
Theorem 2.4 Fekete-Szego Problem. Let the functionfz, given by 2.2, be in the class
Apλ, j, β, a, A, Bandμany complex number. Then
kp2−μk2p1
≤ A−Bδ
p, j1eβ 2aγλrp2, p
×max
⎧ ⎨ ⎩1,
2γ
r λ
p2, pδp2, jaγλrp1, pδp1, jμA−Bδp, j1eβ aγr
λ
p1, p2
⎫ ⎬ ⎭.
2.18
Proof. On using the coefficients ofzp1andzp2, we get
kp1 A−Bβeδ
p, j1 aγλrp1, pδp1, jw1,
kp2
A−Beβδp, j1 aγλrp2, p2δp2, jw2−
A−Beβδp, j1 aγλrp1, pδp1, jw
2 1.
2.19
By using 13that
w2−ρw21≤max
for every complex numberρ, then we can write
kp2−μk2p1
|1
aA−Bδ
p, j1eβ
w2
2γr λ
p2, pδp2, j −
w2 1
γr λ
p1, pδp1, j
− μ
A−Bδp, j1eβ aγλrp1, pδp1, j
2
w12|
| A−Bδ
p, j1eβ 2aγr
λ
p2, pδp2, jw2
−A−Bδ
p, j1eβaγλrp1, pδp1, j−μA−Bδp, j1eβ2
aγλrp1, pδp1, j2 w
2 1|
A−Bδ
p, j1eβ 2aγr
λ
p2, p
w2−hw12,
2.21
where
h 2γ r λ
p2, pδp2, jaγr λ
p1, pδp1, j−μA−Beβδp, j1 aγr
λ
p1, pδp1, j2 . 2.22
3. Integral Operator
Now, we introduce a new integral operator which is denoted by Gη,pz on functions belonging toApas follows:
Gη,pz e
1−ηzpηp
z
ft
t dt
0< η <1, −→0, 3.1
and we verify the effect of this operator on1.11; with a simple calculation, we have
Gη,pz e
1−ηzpηp
⎡ ⎣z
⎛
⎝etp−1 ∞ mp1
kmtm−1
⎞ ⎠dt
⎤
⎦ 0< η <1, −→0
e1−ηzpηezp ∞
mp1
ηp mkmz
m
ezp
∞
mp1
dmzm,
wheredm ηp/mkm. If we putrj 0 in1.11, then we obtain
zfz pfz ≺
aaB A−Bβz
a1Bz 3.3
that is denoted byApλ,0,0, β, a, A, B Apλ, β, a, A, B.
Now if we letAp,ηλ, β, a, A, Bbe a class of functionsGη,pzanalytic inΔand defined by3.1wherefz∈ Apλ, β, a, A, B. Then on using3.1and definition of subordination, we have the following theorem.
Theorem 3.1. Gη,pz∈ Ap,ηλ, β, a, A, Bif and only if
zcgη,p z z2G
η,pz−ep2
1−ηzp pzGη,p −ep2
1−ηzp ≺
aaB A−Bβz
a1Bz . 3.4
Proof. The conditions3.3and3.1give
G
η,pz ep
1−ηzp−1ηpfz
z or fz
zGη,pz ηp −
e1−ηzp
η ,
fz 1 η, p
G
η,pz zGη,p z
−ep
η
1−ηzp−1
1
ηp
Hη,pz
−ep η
1−ηzp−1,
3.5
whereHη,pz zGη,pz. By puttingfzandfzin3.3, we obtain
zfz pfz
zHη,p z−ep21−ηzp Hη,pz−ep2
1−ηzp
zGη,p z z 2G
η,pz−ep2
1−ηzp pzGη,p z−ep2
1−ηzp ≺
aaB A−Bβz a1Bz .
3.6
With a simple calculation onFξz, we have
Fξz
pξ zξ
⎡ ⎣z
0
sξ−1
⎛
⎝esp ∞ mp1
kmsm
⎞ ⎠ds
⎤ ⎦
ezp ∞
mp1
pξ mξkmz
m
ezp
∞
p1
bmzm wherebm pξ mξkm.
LetAp,ξλ, β, a, A, Bbe the class of functionsFξzanalytic inΔdefined byf ∈ Apλ, β, a, A, B. We can write next theorem on using3.3and definition of subordination.
Theorem 3.2. TheFξz∈ Ap,ξλ, β, a, A, Bif and only if
zξ1Fξz zFξz
pξFξz zFξz
≺ a
aB A−Bβz
a1Bz . 3.8
Proof. Since
Fξz
pξ zξ
z
0
sξ−1fsds, 3.9
then we have
fz 1 pξ
ξFξz zFξz
,
fz 1 pξ
ξ1Fξz zFξz.
3.10
Now by making substitutionfzandfzin3.3, we obtain
zfz pfz
z/pξξ1Fξz zFξz
p/pξξFξz zFξz
z
ξ1Fξz zFξz
pξFξz zFξz
≺ a
aB A−Bβz a1Bz .
3.11
Acknowledgments
The authors would like to thank refereesfor the very useful comments that improved the quality of the paper very much. The second author also acknowledges that this research was partially supported by the University Putra Malaysia under the Research University Grant SchemeRUGSno. 05-01-09-0720RU.
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