R E S E A R C H
Open Access
Majorization properties for certain new
classes of analytic functions using the
Salagean operator
Shu-Hai Li
1*, Huo Tang
2,1and En Ao
1*Correspondence:
[email protected] 1School of Mathematics and Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China
Full list of author information is available at the end of the article
Abstract
In the present paper, we investigate the majorization properties for certain classes of multivalent analytic functions defined by the Salagean operator. Moreover, we point out some new and interesting consequences of our main result.
MSC: 30C45
Keywords: analytic functions; multivalent functions;
α
-uniformly starlike functions of orderβ
;α
-uniformly convex functions of orderβ
; subordination; majorization property1 Introduction and definitions
Letf andgbe two analytic functions in the open unit disk
=z∈C:|z|< . (.)
We say thatf is majorized bygin(see []) and write
f(z)g(z) (z∈) (.)
if there exists a functionϕ, analytic in, such that
ϕ(z)≤ and f(z) =ϕ(z)g(z) (z∈). (.)
It may be noted here that (.) is closely related to the concept of quasi-subordination between analytic functions.
For two functionsf andg, analytic in, we say that the functionf is subordinate tog
inif there exists a Schwarz functionω, which is analytic inwith
ω() = and ω(z)< (z∈),
such that
f(z) =gω(z) (z∈).
We denote this subordination byf(z)≺g(z). Furthermore, if the functiong is univalent in, then
f(z)≺g(z) (z∈) ⇔ f() =g() and f()⊂g(). LetApdenote the class of functions of the form
f(z) =zp+
∞
k=p+
akzk
p∈N={, , . . .}, (.)
that are analytic andp-valent in the open unit disk. Also, letA=A.
For a functionf ∈Ap, letf(q)denote aqth-order ordinary differential operator by
f(q)(z) = p! (p–q)!z
p–q+
∞
k=p+
k! (k–q)!akz
k–q, (.)
wherep>q,p∈N,q∈N=N∪{}andz∈. Next, Frasin [] introduced the differential operatorDmf(q)as follows:
Dmf(q)(z) = p!(p–q)
m
(p–q)! z
p–q+
∞
k=p+
k!(k–q)m
(k–q)! akz
k–q. (.)
In view of (.), it is clear thatDf()(z) =f(z),Df()(z) =zf(z) andDmf()(z) =Dmf(z) is
a known operator introduced by Salagean [].
Definition . A functionf(z)∈Ap is said to be in the classLpj,l,q[A,B;α,γ] ofp-valent
functions of complex orderγ = inif and only if
+
γ
Djf(q)(z)
Dlf(q)(z)– (p–q)
j–l
–α
γ
Djf(q)(z)
Dlf(q)(z)– (p–q)
j–l
≺ +Az
+Bz
z∈; –≤B<A≤;j>l;p,j∈N;l,q∈N; ≤α;γ∈C∗=C\ {}. (.)
Clearly, we have the following relationships:
() Ljp,l,q[A,B; ,γ] =Sjp,l,q[A,B;γ];
() Lm,,n[A,B;α, ] =Um,n(α,A,B);
() L,,[ – β, –;α, ] =US(α,β)(≤β< ) (α-uniformly starlike functions of orderβ);
() L,,[ – β, –;α, ] =UK(α,β)(≤β< ) (α-uniformly convex functions of orderβ);
() Lnp+,,n[, –;α,γ] =Sn(p,α,γ)(n∈N);
() L,,[, –;α,γ] =S(α,γ)(≤α< ,γ∈C∗); () L,,[, –;α,γ] =K(α,γ)(≤α< ,γ ∈C∗); () L,,[, –;α, –β] =S∗(α,β)(≤α< ,≤β< ).
The classes Sjp,l,q[A,B;γ] andUm,n(α,A,B) were introduced by Goswami and Aouf []
in [] (see also [–]). The classSn(p, ,γ) =Sn(p,γ) was introduced by Akbulutet al.[].
Also, the classesS(,γ) =S(γ) andK(,γ) =K(γ) are said to be classes of starlike and con-vex of complex orderγ = inwhich were considered by Nasr and Aouf [] and Wia-trowski [] (see also [, ]), andS∗(,β) =S∗(β) denotes the class of starlike functions of orderβin.
A majorization problem for the classS(γ) has recently been investigated by Altintas
et al.[]. Also, majorization problems for the classesS∗(β) andSpj,l,q[A,B;γ] have been
investigated by MacGregor [] and Goswami and Aouf [], respectively. Very recently, Goyal and Goswami [] (see also []) generalized these results for the fractional deriva-tive operator. In the present paper, we investigate a majorization problem for the class
Ljp,l,q[A,B;α,γ].
2 Majorization problem for the classLjp,l,q[A,B;
α,γ
]We begin by proving the following result.
Theorem . Let the function f ∈Ap and suppose that g∈Ljp,l,q[A,B;α,γ].If Djf(q)(z)is
majorized by Dlg(q)(z)in,and
(p–q)j–l≥
(A–B)|γ|
–α + (p–q)
j–l|B|
δ,
then
Dj+f(q)(z)≤Dl+g(q)(z) |z| ≤r
, (.)
where r=r(p,q,α,γ,j,l,A,B)is the smallest positive root of the equation
(A–B)|γ|
–α + (p–q)
j–l|B|
r–(p–q)j–l+ |B|r
–
(A–B)|γ|
–α + (p–q)
j–l|B|+
r+ (p–q)j–l=
–≤B<A≤;p,j∈N;q,l∈N; ≤α< ;γ ∈C∗, ≤δ≤r
. (.)
Proof Suppose thatg∈Ljp,l,q[A,B;α,γ]. Then, making use of the fact that
–α| – | ≺ +Az +Bz ⇔
–αe–iφ+αe–iφ≺ +Az
+Bz (φ∈R),
and letting
= +
γ
Djg(q)(z)
Dlg(q)(z)– (p–q)
j–l
in (.), we obtain
+
γ
Djg(q)(z)
Dlg(q)(z)– (p–q)
or, equivalently,
+
γ
Djg(q)(z)
Dlg(q)(z)– (p–q)
j–l
≺ + (
A–αBe–iφ
–αe–iφ )z
+Bz (.)
which holds true for allz∈. We find from (.) that
+
γ
Djg(q)(z)
Dlg(q)(z)– (p–q)
j–l
= + (
A–αBe–iφ
–αe–iφ )ω(z)
+Bω(z) , (.)
whereω(z) =cz+cz+· · ·,ω∈P,Pdenotes the well-known class of the bounded analytic functions inand satisfies the conditions
ω() = and ω(z)≤ |z| (z∈).
From (.), we get
Djg(q)(z)
Dlg(q)(z)=
(p–q)j–l+ [(A–B)γ
–αe–iφ + (p–q)j–lB]ω(z)
+Bω(z) . (.)
By virtue of (.), we obtain
Dlg(q)(z)≤ +|B||z| (p–q)j–l–|(A–B)γ
–αe–iφ + (p–q)j–lB||z|
Djg(q)(z)
≤ +|B||z|
(p–q)j–l– [(A–B)|γ|
–α + (p–q)j–l|B|]|z|
Djg(q)(z). (.)
Next, sinceDjf(q)(z) is majorized byDlg(q)(z) in, thus from (.), we have
Djf(q)(z) =ϕ(z)Dlg(q)(z).
Differentiating the above equality with respect tozand multiplying byz, we get
Dj+f(q)(z) =zϕ(z)Dlg(q)(z) +ϕ(z)Dl+g(q)(z). (.)
Thus, by noting thatϕ(z)∈Psatisfies the inequality (see,e.g., Nehari []) ϕ(z)≤ –|ϕ(z)|
–|z| (z∈) (.)
and making use of (.) and (.) in (.), we obtain
Dj+f(q)(z)≤ ϕ(z)+ –|ϕ(z)|
–|z| ·
( +|B||z|)|z|
[(p–q)j–l– ((A–B)|γ|
–α + (p–q)j–l|B|)|z|]
which, upon setting
|z|=r and ϕ(z)=ρ (≤ρ≤),
leads us to the inequality
Dj+f(q)(z)
≤ ψ(ρ)
( –r)[(p–q)j–l– ((A–B)|γ|
–α + (p–q)j–l|B|)r]
Dl+g(q)(z),
where
ψ(ρ) = –r +|B|rρ+ –r(p–q)j–l– (A–B)|γ|
–α + (p–q)
j–l|B|
r
ρ
+r +|B|r (.)
takes its maximum value atρ= withr=r(p,q,α,γ,j,l,A,B), where
r=r(p,q,α,γ,j,l,A,B)
is the smallest positive root of equation (.). Furthermore, if ≤δ≤r(p,q,α,γ,j,l,A,B), then the functionψ(ρ) defined by
ψ(ρ) = –δ +|B|δρ+ –δ(p–q)j–l– (A–B)|γ|
–α + (p–q)
j–l|B|
δ
ρ
+δ +|B|δ (.)
is an increasing function on the interval ≤ρ≤ so that
ψ(ρ)≤ψ() = –δ(p–q)j–l– (A–B)|γ|
–α + (p–q)
j–l|B|
δ
≤ρ≤; ≤δ≤r(p,q,α,γ,j,l,A,B). (.)
Hence, upon settingρ= in (.), we conclude that (.) of Theorem . holds true for
|z| ≤r(p,q,α,γ,j,l,A,B), which completes the proof of Theorem ..
Settingα= in Theorem ., we get the following result.
Corollary . Let the function f ∈Apand suppose that g∈Spj,l,q[A,B;γ].If Djf(q)(z)is
ma-jorized by Dlg(q)(z)in,and
(p–q)j–l≥(A–B)|γ|+ (p–q)j–l|B|δ,
then
Dj+f(q)(z)≤Dl+g(q)(z) |z| ≤r
where r=r(p,q,γ,j,l,A,B)is the smallest positive root of the equation
(A–B)|γ|+ (p–q)j–l|B|r–(p–q)j–l+ |B|r–(A–B)|γ|+ (p–q)j–l|B|+ r
+ (p–q)j–l=
–≤B<A≤;p,j∈N;q,l∈N;γ ∈C∗, ≤δ≤r
. (.)
Remark . Corollary . improves the result of Goswami and Aouf [, Theorem ].
Puttingp= ,q= ,j=m,l=n,m>nandγ= in Theorem ., we obtain the following result.
Corollary . Let the function f ∈A and suppose that g∈Um,n(α,A,B).If Dmf(z)is
ma-jorized by Dng(z)in,then
Dm+f(z)≤Dn+g(z) |z| ≤r
, (.)
where r=r(α,A,B)is the smallest positive root of the equation
A–B
–α +|B|
r– + |B|r–
A–B
–α +|B|+
r+ =
(–≤B<A≤; ≤α< ). (.)
ForA= – β,B= –, puttingm= ,n= andm= ,n= in Corollary ., respectively, we obtain the following Corollaries . and ..
Corollary . Let the function f ∈A and suppose that g∈US(α,β).If Df(z)is majorized by g(z)in,then
f(z) +zf(z)≤g(z) |z| ≤r
,
where r=r(α,β)is the smallest positive root of the equation
( –β) –α +
r– r–
( –β) –α +
r+ = (≤α< ; ≤β< ).
Corollary . Let the function f ∈A and suppose that g∈UK(α,β).If Df(z)is majorized
by Dg(z)in,then
Df(z)≤Dg(z) |z| ≤r
,
where r=r(α,β)is the smallest positive root of the equation
( –β) –α +
r– r–
( –β) –α +
r+ = (≤α< ; ≤β< ).
Corollary . Let the function f ∈Apand suppose that g∈Sn(p,α,γ).If Dn+f(z)is
ma-jorized by Dng(z)in,then
Dn+f(z)≤Dn+g(z) |z| ≤r
, (.)
where r=r(p,α,γ)is the smallest positive root of the equation
|γ|
–α+p
r– (p+ )r–
|γ|
–α +p+
r+p=
p∈N;γ ∈C∗; ≤α< . (.)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Author details
1School of Mathematics and Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China.2School of
Mathematical Sciences, Beijing Normal University, Beijing, 100875, China.
Acknowledgements
Dedicated to Professor Hari M. Srivastava.
The present investigation is partly supported by the Natural Science Foundation of Inner Mongolia of People’s Republic of China under Grant 2009MS0113, 2010MS0117. The authors would like to thank the referees for their helpful comments and suggestions to improve our manuscript.
Received: 9 November 2012 Accepted: 14 February 2013 Published: 4 March 2013 References
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doi:10.1186/1029-242X-2013-86