APPLICATIONS OF CERTAIN FUNCTIONS ASSOCIATED WITH LEMNISCATE BERNOULLI | Halim | Journal of the Indonesian Mathematical Society 115 317 2 PB

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J. Indones. Math. Soc.

Vol. 18, No. 2 (2012), pp. 93–99.

APPLICATIONS OF CERTAIN FUNCTIONS

ASSOCIATED WITH LEMNISCATE BERNOULLI

Suzeini Abdul Halim

1

and Rashidah Omar

2

1,2_{Institute of Mathematical Sciences,}

Faculty of Science, University of Malaya, Malaysia, [email protected]

2_{Faculty of Computer and Mathematical Sciences,}

MARA University of Technology Malaysia,

[email protected]

Abstract.ForJ(α, f(z)) = (1−α)zf_f₍′_z(z₎)+α[1 +zf_f′′′((zz))

]

(α_≥0), denoteSL(α) and SLc _{as classes of} _α_{-convex and convex functions which respectively satisfy}

conditions|[J(α, f(z))]2−1|<1 and

[

1 +zf_f′′′((zz))

]2 −1

<1. Using established

results, namely 1 +βzp′₍_z₎_≺ 1+Dz

1+Ez ,1 + βzp′₍z)

p(z) ≺ 1+Dz

1+Ez and 1 + βzp′₍z)

p2(z) ≺ 1+Dz

1+Ez

implyp(z)≺√1 +z wherep(z) is an analytic function defined on the open unit diskDwithp(0) = 1. This article obtains conditions so that analytic functionsf

belong to the classesSL(α) andSLc_.

Key words: Convex functions, differential subordination, lemniscate Bernoulli.

Abstrak. Untuk J(α, f(z)) = (1−α)zf_f′₍(_zz₎) +α [

1 +zf_f′′′((zz))

]

(α ≥ 0), ny-atakanSL(α) dan SLc _{sebagai kelas-kelas dari konveks-}_α _{and fungsi-fungsi}

kon-veks yang secaraberturut-turut memenuhi kondisi |[J(α, f(z))]2 −1| < 1 dan

[

1 +zf_f′′′((zz))

]2 −1

<1. Dengan menggunakan hasil-hasil yang telah diperoleh

se-belumnya, yaitu 1 +βzp′₍_z₎_≺1+Dz

1+Ez ,1 + βzp′₍_z₎

p(z) ≺ 1+Dz

1+Ez dan 1 + βzp′₍_z₎

p2(z) ≺ 1+Dz

1+Ez

mengakibatkan p(z) ≺ √1 +z dimana p(z) adalah sebuah fungsi analitis yang didefinisikan pada cakram bukaDdenganp(0) = 1. Artikel ini memperoleh kondisi-kondisi sehingga fungsi-fungsi analitisf berada dalam kelas-kelasSL(α) danSLc_.

Kata kunci: Fungsi Konveks, subordinasi diferensial, lemniscate Bernoulli.

2000 Mathematics Subject Classification: 30C45, 30C50. Received: 25-10-2011, revised: 25-07-2012, accepted: 26-07-2012.

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1. Introduction

LetAdenote the class of all analytic functionsf in the open unit diskD:=

{z∈C:|z|<1} and normalised byf(0) = 0, f′(0) = 1. An analytic functionf is subordinate to an analytic function g, writtenf(z)≺g(z)(z∈D), if there exists an analytic function w in D such that w(0) = 0 and |w(z)| < 1 for |z| < 1 and f(z) =g(w(z)). In particular, ifgis univalent inD, thenf(z)≺g(z) is equivalent tof(0) =g(0) andf(D)⊂g(D). Forf inAandz∈D, the classes of starlikeS⋆, convexCandα-convex functions are defined respectively by the conditions

S⋆= {

z∈D:Rezf ′₍_z₎

f(z) >0 }

C= {

z∈D:Rezf ′′₍_z₎

f′₍_z₎ >−1

}

Mα={z∈D:Re[J(α, f(z))]>0} whereJ(α, f(z)) = (1−α)zf_f(z)′(z)+α[1 + zf_f′′′_(z)(z)

]

, α≥0. Note thatM0=S⋆and M1=C.

Sok´ol and Stankiewicz [5] introduced the class SL⋆ _{consisting of normalised}

analytic functions f in D satisfying the condition [

zf′_(z)

f(z) ]2

−1

<1 for z ∈ D.

Geometrically, a function f _∈ SL⋆ _if zf′_(z)

f(z) is in the interior of the right half of the lemniscate of Bernoulli (x2+y2)2₋₂₍_x2₋_y2_{) = 0. A function in the class} SL⋆_{is called a Sok´}_o_{l-Stankiewicz starlike function. Alternatively, we can also write} f _∈SL⋆ _⇔ zf′_(z)

f(z) ≺ √

1 +z .Some properties of functions in classSL⋆ _{have been} studied by [1], [4], [6], [7], [8] and [9]. Next, we denote S⋆[A, B] as the class of Janowski starlike functions defined in Janowski [2] consisting of functions f _{∈ A} satisfying zf_f(z)′(z)≺ 1+Az

1+Bz (−1≤B < A≤1).

Let SL(α) and SLc denote the classes of α-convex and convex functions

which respectively satisfy|[J(α, f(z))]2−1|<1 and [

1 + zf_f′′′_(z)(z)

]2 −1

<1 (z∈

D). It is obvious that f _{∈ SL}(α) ⇔ J(α, f(z)) ≺ √1 +z and f _{∈ SL}c _⇔ 1 + zf_f′′′(z)(z) ≺

√

1 +z . Recently in [4] the authors determined condition on β so

that 1 +βzp′₍_z₎_,_{1 +}βzp′_(z)

p(z) and 1 + βzp′_(z)

p2(z) are subordinated to 1+Dz

1+Ez imply p(z) is subordinated to √1 +z where p(z) is analytic inD with p(0) = 1. Properties of functions in the classSL(α) andSLc are obtained using results given in [4]. In proving the lemmas, the following proposition is needed.

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Applications of Certain Functions 95

With appropriate choices of ϕ in Proposition 1.1, and using similar approach as above Lemma 1.3 and Lemma 1.4 is easily verified. Details of proving these lemmas can be found in [4].

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Corollary 2.2. Let β0 = 2√2(₁_−||D_E−_|E|) , |E| < 1 , |D| ≤ 1 , D ̸= E , β ≥ β0

andf _∈A. Iff satisfies

1 +βzf

′′₍_z₎

f′₍_z₎

{

z[f′′(z)]′

f′′₍_z₎ −

zf′′(z) f′₍_z₎ + 1

}

≺1 +Dz

1 +Ez

thenf _{∈ SL}c.

Proof. _With_p₍_z_{) = 1 +}zf′′(z)

f′_(z) , it follows that

zp′(z) =z

2_[_f′′₍_z_)]′

f′₍_z₎ +

zf′′(z) f′₍_z₎ −

z2[f′′(z)]2 [f′₍_z_)]2

=zf

′′₍_z₎

f′₍_z₎

{

z[f′′(z)]′

f′′₍_z₎ + 1−

zf′′(z) f′₍_z₎

} ,

and applying Lemma 1.2 gives 1 +zf_f′′′(z)(z) ≺ 1+Dz

1+Ez, hencef ∈ SL c_.

Theorem 2.3. Let β0= ₍₁4|D_−|−_EE_|₎| andβ_≥β0. Suppose f _∈Aand

1+β    

  

(1−α)zf′₍_z₎

[

1−zff(z)′(z)+ zf′′(z)

f′(z) ]

(1−α)zf′₍_z_{) +}_αf₍_z₎[_{1 +} zf′′(z)

f′(z) ] +

αzf′′(z)[1−zff′′′(z)(z)+ z[f′′_(z)]′

f′′(z) ]

(1−α)f′₍_z₎zf′_(z)

f(z) +α[f′(z) +zf′′(z)]    

  

≺1 +_{1 +}Dz Ez

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and based on Lemma 1.3,f ∈ SLc_.

Theorem 2.5. Let β0= 4√₍₁2_−||D_E−_|₎E| , β≥β0 andf ∈A.

1+βz    

  

(1−α)[f′(z)]3_f₍_z₎ [

zf′′(z) f′_(z) + 1−

zf′_(z)

f(z) ]

+α[f(z)]2_f′₍_z₎_f′′₍_z₎[z[f′′_(z)]′

f′′_(z) + 1−

zf′′_(z)

f′_(z)

]

[(1−α)f′₍_z₎_zf′₍_z_{) +}_αf₍_z_)[_f′₍_z_{) +}_zf′′₍_z_)]]2

   

  

≺1 +_{1 +}Dz

Ez ⇒f ∈ SL(α).

Proof. _{Using Lemma 1.4 with}_p₍_z_{) = (1}₋_α₎[zf′(z)

f(z) ]

+α[1 +zf_f′′′_(z)(z)

]

gives the desired result.

Corollary 2.6. Let β0= 4

√

2|D−E|

(1−|E|) , β ≥β0 andf ∈A.

1 +β zf

′₍_z₎_f′′₍_z₎

[f′₍_z_{) +}_zf′′₍_z_)]2

{_z_[_f_′′₍_z_)]_′

f′′₍_z₎ + 1−

zf′′₍_z₎

f′₍_z₎

}

≺ 1 +_{1 +}Dz

Ez ⇒f ∈ SL c

.

Proof. _Set_p₍_z_{) = 1 +}zf′′(z)

f′(z) and in a similar manner the result is easily obtained.

3. Concluding Remark

Forα= 0, Theorem 2.1, Theorem 2.3 and Theorem 2.5 reduce to the results forf _∈SL⋆_.

AcknowledgementThis research was supported by University of Malaya grants: UMRG108/10APR.

References

[1] Ali, R.M., Chu, N.E., Ravichandran, V. and Sivaprasad Kumar, S. First order differential subordination for functions associated with the lemniscate of Bernoulli,Taiwanese Journal of Mathematics,16_{(3) (2012) 1017-1026.}

[2] Janowski, W., Some extremal problems for certain families of analytic functions I, Ann. Polon. Math.,28_{(1973) 297-326.}

[3] Miller, S.S. and Mocanu, P.T., Differential subordination, theory and application, Marcel Dekker, Inc., New York, Basel, 2000.

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Applications of Certain Functions 99

[5] Sok´ol, J. and Stankiewicz, J. Radius of convexity of some subclasses of strongly starlike functions,Folia Scient. Univ. Tech. Resoviensis,19_{(1996) 101-105.}

[6] Sok´ol, J., Radius problems in the classSL⋆_,_{Applied Mathematics and Computation}_,₂₁₄

(2009) 569-573.

[7] Sok´ol, J., Coefficient estimates in a class of strongly starlike functions,Kyungpook Math. J.,

49(2009) 349-353.

[8] Sok´ol, J., On application of certain sufficient condition for starlikeness,J. Math. Appl.,30

(2008) 46-53.