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Majorization properties for certain new classes of analytic functions using the Salagean operator

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.

1 Introduction and definitions

Let f and g be two analytic functions in the open unit disk

Δ= { z C : | z | < 1 } .
(1.1)

We say that f is majorized by g in Δ (see [1]) and write

f(z)g(z)(zΔ)
(1.2)

if there exists a function φ, analytic in Δ, such that

| φ ( z ) | 1andf(z)=φ(z)g(z)(zΔ).
(1.3)

It may be noted here that (1.2) is closely related to the concept of quasi-subordination between analytic functions.

For two functions f and g, analytic in Δ, we say that the function f is subordinate to g in Δ if there exists a Schwarz function ω, which is analytic in Δ with

ω(0)=0and | ω ( z ) | <1(zΔ),

such that

f(z)=g ( ω ( z ) ) (zΔ).

We denote this subordination by f(z)g(z). Furthermore, if the function g is univalent in Δ, then

f(z)g(z)(zΔ)f(0)=g(0)andf(Δ)g(Δ).

Let A p denote the class of functions of the form

f(z)= z p + k = p + 1 a k z k ( p N = { 1 , 2 , } ) ,
(1.4)

that are analytic and p-valent in the open unit disk Δ. Also, let A 1 =A.

For a function f A p , let f ( q ) denote a q th-order ordinary differential operator by

f ( q ) (z)= p ! ( p q ) ! z p q + k = p + 1 k ! ( k q ) ! a k z k q ,
(1.5)

where p>q, pN, q N 0 =N{0} and zΔ. Next, Frasin [2] introduced the differential operator D m f ( q ) as follows:

D m f ( q ) (z)= p ! ( p q ) m ( p q ) ! z p q + k = p + 1 k ! ( k q ) m ( k q ) ! a k z k q .
(1.6)

In view of (1.6), it is clear that D 0 f ( 0 ) (z)=f(z), D 0 f ( 1 ) (z)=z f (z) and D m f ( 0 ) (z)= D m f(z) is a known operator introduced by Salagean [3].

Definition 1.1 A function f(z) A p is said to be in the class L p , q j , l [A,B;α,γ] of p-valent functions of complex order γ0 in Δ if and only if

(1.7)

Clearly, we have the following relationships:

  1. (1)

    L p , q j , l [A,B;0,γ]= S p , q j , l [A,B;γ];

  2. (2)

    L 1 , 0 m , n [A,B;α,1]= U m , n (α,A,B);

  3. (3)

    L 1 , 0 1 , 0 [12β,1;α,1]=US(α,β) (0β<1) (α-uniformly starlike functions of order β);

  4. (4)

    L 2 , 1 1 , 0 [12β,1;α,1]=UK(α,β) (0β<1) (α-uniformly convex functions of order β);

  5. (5)

    L p , 0 n + 1 , n [1,1;α,γ]= S n (p,α,γ) (n N 0 );

  6. (6)

    L 1 , 0 1 , 0 [1,1;α,γ]=S(α,γ) (0α<1, γ C );

  7. (7)

    L 1 , 0 2 , 1 [1,1;α,γ]=K(α,γ) (0α<1, γ C );

  8. (8)

    L 1 , 0 1 , 0 [1,1;α,1β]= S (α,β) (0α<1, 0β<1).

The classes S p , q j , l [A,B;γ] and U m , n (α,A,B) were introduced by Goswami and Aouf [4] and Li and Tang [5], respectively. The classes US(α,β) and UK(α,β) were studied recently in [6] (see also [712]). The class S n (p,0,γ)= S n (p,γ) was introduced by Akbulut et al. [13]. Also, the classes S(0,γ)=S(γ) and K(0,γ)=K(γ) are said to be classes of starlike and convex of complex order γ0 in Δ which were considered by Nasr and Aouf [14] and Wiatrowski [15] (see also [16, 17]), and S (0,β)= S (β) denotes the class of starlike functions of order β in Δ.

A majorization problem for the class S(γ) has recently been investigated by Altintas et al. [18]. Also, majorization problems for the classes S (β) and S p , q j , l [A,B;γ] have been investigated by MacGregor [1] and Goswami and Aouf [4], respectively. Very recently, Goyal and Goswami [19] (see also [20]) generalized these results for the fractional derivative operator. In the present paper, we investigate a majorization problem for the class L p , q j , l [A,B;α,γ].

2 Majorization problem for the class L p , q j , l [A,B;α,γ]

We begin by proving the following result.

Theorem 2.1 Let the function f A p and suppose that g L p , q j , l [A,B;α,γ]. If D j f ( q ) (z) is majorized by D l g ( q ) (z) in Δ, and

( p q ) j l [ ( A B ) | γ | 1 α + ( p q ) j l | B | ] δ,

then

| D j + 1 f ( q ) (z)|| D l + 1 g ( q ) (z)| ( | z | r 0 ) ,
(2.1)

where r 0 = r 0 (p,q,α,γ,j,l,A,B) is the smallest positive root of the equation

(2.2)

Proof Suppose that g L p , q j , l [A,B;α,γ]. Then, making use of the fact that

ϖα|ϖ1| 1 + A z 1 + B z ϖ ( 1 α e i ϕ ) +α e i ϕ 1 + A z 1 + B z (ϕR),

and letting

ϖ=1+ 1 γ ( D j g ( q ) ( z ) D l g ( q ) ( z ) ( p q ) j l )

in (1.7), we obtain

[ 1 + 1 γ ( D j g ( q ) ( z ) D l g ( q ) ( z ) ( p q ) j l ) ] ( 1 α e i ϕ ) +α e i ϕ 1 + A z 1 + B z

or, equivalently,

1+ 1 γ ( D j g ( q ) ( z ) D l g ( q ) ( z ) ( p q ) j l ) 1 + ( A α B e i ϕ 1 α e i ϕ ) z 1 + B z
(2.3)

which holds true for all zΔ.

We find from (2.3) that

1+ 1 γ ( D j g ( q ) ( z ) D l g ( q ) ( z ) ( p q ) j l ) = 1 + ( A α B e i ϕ 1 α e i ϕ ) ω ( z ) 1 + B ω ( z ) ,
(2.4)

where ω(z)= c 1 z+ c 2 z 2 + , ωP, P denotes the well-known class of the bounded analytic functions in Δ and satisfies the conditions

ω(0)=0and | ω ( z ) | |z|(zΔ).

From (2.4), we get

D j g ( q ) ( z ) D l g ( q ) ( z ) = ( p q ) j l + [ ( A B ) γ 1 α e i ϕ + ( p q ) j l B ] ω ( z ) 1 + B ω ( z ) .
(2.5)

By virtue of (2.5), we obtain

| D l g ( q ) ( z ) | 1 + | B | | z | ( p q ) j l | ( A B ) γ 1 α e i ϕ + ( p q ) j l B | | z | | D j g ( q ) ( z ) | 1 + | B | | z | ( p q ) j l [ ( A B ) | γ | 1 α + ( p q ) j l | B | ] | z | | D j g ( q ) ( z ) | .
(2.6)

Next, since D j f ( q ) (z) is majorized by D l g ( q ) (z) in Δ, thus from (1.3), we have

D j f ( q ) (z)=φ(z) D l g ( q ) (z).

Differentiating the above equality with respect to z and multiplying by z, we get

D j + 1 f ( q ) (z)=z φ (z) D l g ( q ) (z)+φ(z) D l + 1 g ( q ) (z).
(2.7)

Thus, by noting that φ(z)P satisfies the inequality (see, e.g., Nehari [21])

| φ ( z ) | 1 | φ ( z ) | 2 1 | z | 2 (zΔ)
(2.8)

and making use of (2.6) and (2.8) in (2.7), we obtain

| D j + 1 f ( q ) ( z ) | ( | φ ( z ) | + 1 | φ ( z ) | 2 1 | z | 2 ( 1 + | B | | z | ) | z | [ ( p q ) j l ( ( A B ) | γ | 1 α + ( p q ) j l | B | ) | z | ] ) × | D l + 1 g ( q ) ( z ) | ,
(2.9)

which, upon setting

|z|=rand | φ ( z ) | =ρ(0ρ1),

leads us to the inequality

where

ψ ( ρ ) = r ( 1 + | B | r ) ρ 2 + ( 1 r 2 ) [ ( p q ) j l ( ( A B ) | γ | 1 α + ( p q ) j l | B | ) r ] ρ + r ( 1 + | B | r )
(2.10)

takes its maximum value at ρ=1 with r 0 = r 0 (p,q,α,γ,j,l,A,B), where

r 0 = r 0 (p,q,α,γ,j,l,A,B)

is the smallest positive root of equation (2.2). Furthermore, if 0δ r 0 (p,q,α,γ,j,l,A,B), then the function ψ(ρ) defined by

ψ ( ρ ) = δ ( 1 + | B | δ ) ρ 2 + ( 1 δ 2 ) [ ( p q ) j l ( ( A B ) | γ | 1 α + ( p q ) j l | B | ) δ ] ρ + δ ( 1 + | B | δ )
(2.11)

is an increasing function on the interval 0ρ1 so that

(2.12)

Hence, upon setting ρ=1 in (2.11), we conclude that (2.1) of Theorem 2.1 holds true for |z| r 0 (p,q,α,γ,j,l,A,B), which completes the proof of Theorem 2.1. □

Setting α=0 in Theorem 2.1, we get the following result.

Corollary 2.1 Let the function f A p and suppose that g S p , q j , l [A,B;γ]. If D j f ( q ) (z) is majorized by D l g ( q ) (z) in Δ, and

( p q ) j l [ ( A B ) | γ | + ( p q ) j l | B | ] δ,

then

| D j + 1 f ( q ) (z)|| D l + 1 g ( q ) (z)| ( | z | r 0 ) ,
(2.13)

where r 0 = r 0 (p,q,γ,j,l,A,B) is the smallest positive root of the equation

(2.14)

Remark 2.1 Corollary 2.1 improves the result of Goswami and Aouf [[4], Theorem 1].

Putting p=1, q=0, j=m, l=n, m>n and γ=1 in Theorem 2.1, we obtain the following result.

Corollary 2.2 Let the function fA and suppose that g U m , n (α,A,B). If D m f(z) is majorized by D n g(z) in Δ, then

| D m + 1 f(z)|| D n + 1 g(z)| ( | z | r 0 ) ,
(2.15)

where r 0 = r 0 (α,A,B) is the smallest positive root of the equation

(2.16)

For A=12β, B=1, putting m=1, n=0 and m=2, n=1 in Corollary 2.2, respectively, we obtain the following Corollaries 2.3 and 2.4.

Corollary 2.3 Let the function fA and suppose that gUS(α,β). If Df(z) is majorized by g(z) in Δ, then

| f (z)+z f (z)|| g (z)| ( | z | r 0 ) ,

where r 0 = r 0 (α,β) is the smallest positive root of the equation

[ 2 ( 1 β ) 1 α + 1 ] r 3 3 r 2 [ 2 ( 1 β ) 1 α + 3 ] r+1=0(0α<1;0β<1).

Corollary 2.4 Let the function fA and suppose that gUK(α,β). If D 2 f(z) is majorized by Dg(z) in Δ, then

| D 3 f(z)|| D 2 g(z)| ( | z | r 0 ) ,

where r 0 = r 0 (α,β) is the smallest positive root of the equation

[ 2 ( 1 β ) 1 α + 1 ] r 3 3 r 2 [ 2 ( 1 β ) 1 α + 3 ] r+1=0(0α<1;0β<1).

Also, putting A=1, B=1, q=0, j=n+1 and l=n in Theorem 2.1, we obtain the following result.

Corollary 2.5 Let the function f A p and suppose that g S n (p,α,γ). If D n + 1 f(z) is majorized by D n g(z) in Δ, then

| D n + 2 f(z)|| D n + 1 g(z)| ( | z | r 0 ) ,
(2.17)

where r 0 = r 0 (p,α,γ) is the smallest positive root of the equation

(2.18)

References

  1. 1.

    MacGregor TH: Majorization by univalent functions. Duke Math. J. 1967, 34: 95–102. 10.1215/S0012-7094-67-03411-4

    MathSciNet  Article  Google Scholar 

  2. 2.

    Frasin BA: Neighborhoods of certain multivalent functions with negative coefficients. Appl. Math. Comput. 2007, 193: 1–6. 10.1016/j.amc.2007.03.026

    MathSciNet  Article  Google Scholar 

  3. 3.

    Salagean GS: Subclasses of univalent functions. Lecture Notes in Math. 1013. In Complex Analysis - Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981). Springer, Berlin; 1983:362–372.

    Google Scholar 

  4. 4.

    Goswami P, Aouf MK: Majorization properties for certain classes of analytic functions using the Salagean operator. Appl. Math. Lett. 2010, 23: 1351–1354. 10.1016/j.aml.2010.06.030

    MathSciNet  Article  Google Scholar 

  5. 5.

    Li S-H, Tang H: Certain new classes of analytic functions defined by using the Salagean operator. Bull. Math. Anal. Appl. 2010, 4(2):62–75.

    Google Scholar 

  6. 6.

    Kanas S, Srivastava HM: Linear operators associated with k -uniformly convex functions. Integral Transforms Spec. Funct. 2000, 9: 121–132. 10.1080/10652460008819249

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kanas S, Yaguchi T: Subclasses of k -uniformly convex and starlike functions defined by generalized derivative, I. Indian J. Pure Appl. Math. 2001, 32(9):1275–1282.

    MathSciNet  Google Scholar 

  8. 8.

    Kanas S: Integral operators in classes k -uniformly convex and k -starlike functions. Mathematica (Cluj-Napoca, 1992) 2001, 43(66)(1):77–87.

    MathSciNet  Google Scholar 

  9. 9.

    Kanas S, Wiśniowska A: Conic regions and k -uniform convexity, II. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 1998, 170: 65–78.

    Google Scholar 

  10. 10.

    Kanas S: Differential subordination related to conic sections. J. Math. Anal. Appl. 2006, 317(2):650–658. 10.1016/j.jmaa.2005.09.034

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ramachandran C, Srivastava HM, Swaminathan A: A unified class of k -uniformly convex functions defined by the Salagean derivative operator. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 2007, 55: 47–59.

    MathSciNet  Google Scholar 

  12. 12.

    Shams S, Kulkarni SR, Jahangiri JM: On a class of univalent functions defined by Ruscheweyh derivatives. Kyungpook Math. J. 2003, 43: 579–585.

    MathSciNet  Google Scholar 

  13. 13.

    Akbulut S, Kadioglu E, Ozdemir M: On the subclass of p -valently functions. Appl. Math. Comput. 2004, 147(1):89–96. 10.1016/S0096-3003(02)00653-7

    MathSciNet  Article  Google Scholar 

  14. 14.

    Nasr MA, Aouf MK: Starlike function of complex order. J. Nat. Sci. Math. 1985, 25(1):1–12.

    MathSciNet  Google Scholar 

  15. 15.

    Wiatrowski P: On the coefficients of some family of holomorphic functions. Zeszyry Nauk. Univ. Lodz. Nauk. Mat.-Przyrod. Ser. II 1970, 39: 75–85.

    Google Scholar 

  16. 16.

    Kanas S, Darwish HE: Fekete-Szego problem for starlike and convex functions of complex order. Appl. Math. Lett. 2010, 23: 777–782. 10.1016/j.aml.2010.03.008

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kanas S, Sugawa T: On conformal representation of the interior of an ellipse. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 2006, 31: 329–348.

    MathSciNet  Google Scholar 

  18. 18.

    Altintas O, Ozkan O, Srivastava HM: Majorization by starlike functions of complex order. Complex Var. Theory Appl. 2001, 46: 207–218. 10.1080/17476930108815409

    MathSciNet  Article  Google Scholar 

  19. 19.

    Goyal SP, Goswami P: Majorization for certain classes of analytic functions defined by fractional derivatives. Appl. Math. Lett. 2009, 22(12):1855–1858. 10.1016/j.aml.2009.07.009

    MathSciNet  Article  Google Scholar 

  20. 20.

    Prajapat JK, Aouf MK: Majorization problem for certain class of p -valently analytic function defined by generalized fractional differential operator. Comput. Math. Appl. 2012, 63: 42–47.

    MathSciNet  Google Scholar 

  21. 21.

    Nehari Z: Conformal Mapping. McGraw-Hill, New York; 1955.

    Google Scholar 

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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.

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Correspondence to Shu-Hai Li.

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Li, S., Tang, H. & Ao, E. Majorization properties for certain new classes of analytic functions using the Salagean operator. J Inequal Appl 2013, 86 (2013). https://doi.org/10.1186/1029-242X-2013-86

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Keywords

  • analytic functions
  • multivalent functions
  • α-uniformly starlike functions of order β
  • α-uniformly convex functions of order β
  • subordination
  • majorization property