In this paper we consider the Ln W A ! A, Lnf .´/D.1 /Dnf .´/CInf .´/linear operator, whereDnis the Sˇalˇagean differential operator andInis the Sˇalˇagean integral operator

Vol. 19 (2018), No. 2, pp. 1095–1106 DOI: 10.18514/MMN.2018.2457

ON A CLASS OF UNIVALENT FUNCTIONS DEFINED BY S ˇAL ˇAGEAN INTEGRO-DIFFERENTIAL OPERATOR

A. O. P ´´ ALL-SZAB ´O Received 22 November, 2017

Abstract. In this paper we consider the Lⁿ W A ! A,

Lⁿf .´/D.1 /Dⁿf .´/CIⁿf .´/linear operator, whereDⁿis the Sˇalˇagean differential operator andIⁿis the Sˇalˇagean integral operator. We study several differential subordinations generated byLⁿ. We introduce a class of holomorphic functionsL^m_n.ˇ/, and obtain some subordination results.

2010Mathematics Subject Classification: 30C45; 30A20; 34A40

Keywords: analytic functions, convex function, Sˇalˇagean integro-differential operator, differen- tial operator, differential subordination, dominant, best dominant

1. PRELIMINARIES

LetU be the unit disk in the complex plane:

U D f´2CW j´j< 1g:

LetH.U /be the space of holomorphic functions inU and let AmD˚

f 2H.U /W f .´/D´CamC1´^mC1C ; ´2U withA1DA. Fora2Candm2N,N0DN[ f0g;ND f1; 2; : : :glet

HŒa; mD˚

f 2H.U /W f .´/DaCam´^mCamC1´^mC1C ; ´2U : Denote by

KD

f 2AW <´f⁰⁰.´/

f⁰.´/ C1 > 0; ´2U

the class of normalized convex functions inU.

Definition 1([5], def. 3.5.1). Letf andgbe analytic functions inU. We say that the functionf is subordinate to the functiong, if there exists a functionw, which is analytic inU andw.0/D0I jw.´/j< 1I´2U, such thatf .´/Dg.w.´//I 8´2U:

We denote bythe subordination relation. Ifgis univalent, thenf gif and only iff .0/Dg.0/andf .U /g .U /.

c 2018 Miskolc University Press

1096 A. O. P ´ALL-SZAB ´O

Let WC³U !Cbe a function and lethbe univalent inU. Ifpis analytic in U and satisfies the (second-order) differential subordination

.i / p .´/ ; ´p⁰.´/ ; ´²p⁰⁰.´/I´

h .´/ ; .´2U /

thenpis called a solution of the differential subordination. The univalent functionq is called a dominant of the solution of the differential subordination, or more simply a dominant, ifp q for allp satisfying.i /. A dominanteq, which satisfieseqq for all dominantsq of.i /is said to be the best dominant of.i /. The best dominant is unique up to a rotation of U. In order to prove the original results we use the following lemmas.

Lemma 1 (Hallenbeck and Ruscheweyh, [2]). Leth be a convex function with h.0/Da, and let2Cbe a complex number with<0. Ifp2HŒa; nand

p.´/C1

´p⁰.´/h.´/; ´2U then

p.´/q.´/h.´/; ´2U where

q.´/D n´⁼ⁿ

Z ´ 0

h.t /t^{=n 1}dt; ´2U:

Lemma 2(Miller and Mocanu, [3]). Letqbe a convex function inU and let h.´/Dq.´/Cn˛´q⁰.´/; ´2U

where˛ > 0andnis a positive integer. If

p.´/Dq.0/Cpn´ⁿCpnC1´^nC1C ; ´2U is holomorphic inU and

p.´/Cn˛´p⁰.´/h.´/; ´2U then

p.´/q.´/

and this result is sharp.

Definition 2([8]). For f 2A; n2N0, the Sˇalˇagean differential operator Dⁿ is defined byDⁿWA!A,

D⁰f .´/Df .´/;

: : :

D^nC1f .´/D´ Dⁿf .´/₀

; ´2U

Remark1. Iff 2Aandf .´/D´C

1

X

kD2

a_k´^k, then

Dⁿf .´/D´C

1

X

kD2

kⁿa_k´^k; ´2U:

Definition 3([8]). Forf 2A; n2N0DN[ f0g;ND f1; 2; : : :g, the operatorIⁿ is defined by

I⁰f .´/Df .´/;

: : :

Iⁿf .´/DI I^{n 1}f .´/

; ´2U Remark2. Iff 2Aandf .´/D´C

1

X

kD2

a_k´^k, then

Iⁿf .´/D´C

1

X

kD2

a_k kⁿ´^k;

´2U,.n2N0/and´ .Iⁿf .´//⁰DI^{n 1}f .´/.

Definition 4. Let0; n2N. Denote byLⁿthe operator given by LⁿWA!A,

Lⁿf .´/D.1 /Dⁿf .´/CIⁿf .´/ ; ´2U:

Remark3. Iff 2Aandf .´/D´C

1

X

kD2

a_k´^k, then

Lⁿf .´/D´C

1

X

kD2

kⁿ.1 /C 1 kⁿ

a_k´^k; ´2U: (1.1) 2. MAIN RESULTS

Theorem 1. Letqbe a convex function,q.0/D1and lethbe the function h.´/Dq.´/C´q⁰.´/; ´2U:

Iff 2A,0,n2Nand satisfies the differential subordination Lⁿf .´/0

h.´/; ´2U (2.1)

then Lⁿf .´/

´ q.´/; ´2U and this result is sharp.

1098 A. O. P ´ALL-SZAB ´O

Proof. Let

p.´/DLⁿf .´/

´ D

´C

1

X

kD2

kⁿ.1 /C 1 kⁿ

a_k´^k

´ D1Cpn´ⁿCpnC1´^nC1C (2.2)

´2U. From (2.2) we havep2HŒ1; 1. Let

Lⁿf .´/D´p.´/; ´2U: (2.3) Differentiating (2.3), we obtain

Lⁿf .´/0

Dp.´/C´p⁰.´/; ´2U: (2.4) Then (2.1) becomes

p.´/C´p⁰.´/h.´/; ´2U: (2.5) By using Lemma2, we have

p.´/q.´/; ´2U;

i.e. Lⁿf .´/

´ q.´/; ´2U:

Remark4. IfD0we get Theorem 4 from Oros [6] and forD1we get Theorem 4 from Bˇalˇaet¸i [1].

Example1. ForD0,nD1,f 2Awe deduce that f⁰.´/C´f⁰⁰.´/ 1

.1 ´/²; ´2U implies

f⁰.´/ 1

1 ´; ´2U:

Example2. ForD1,nD1,f 2Awe deduce that f .´/

´ 1

.1 ´/²; ´2U implies

R´

0 f .t / t ¹dt

´ 1

1 ´; ´2U:

Theorem 2. Letqbe a convex function,q.0/D1and lethbe the function h.´/Dq.´/C´q⁰.´/; ´2U:

Iff 2A,0,n2Nand satisfies the differential subordination ´L^nC1f .´/

Lⁿf .´/

0

h.´/; ´2U (2.6)

then L^nC1f .´/

Lⁿf .´/ q.´/; ´2U and this result is sharp.

Proof. Let

p.´/D L^nC1f .´/

Lⁿf .´/ D

´C

1

X

kD2

k^nC1.1 /C 1 k^nC1

a_k´^k

´C

1

X

kD2

kⁿ.1 /C 1 kⁿ

a_k´^k

:

We havep⁰.´/D L^nC1f .´/0

Lⁿf .´/ p.´/ Lⁿf .´/0

Lⁿf .´/ and p.´/C´p⁰.´/D

´L^nC1f .´/

Lⁿf .´/

0

. Relation (2.6) becomes

p.´/C´p⁰.´/h.´/Dq.´/C´q⁰.´/; ´2U:

By using Lemma2we have

p.´/q.´/ i:e: L^nC1f .´/

Lⁿf .´/ q.´/; ´2U:

Theorem 3. Letqbe a convex function,q.0/D1and lethbe the function

h.´/Dq.´/C´q⁰.´/; ´2U:

Iff 2A,0,n2Nand satisfies the differential subordination L^nC1f .´/0

Ch

I^{n 1}f .´/0

I^nC1f .´/0i

h.´/; ´2U (2.7) then

Lⁿf .´/₀

q.´/; ´2U and this result is sharp.

1100 A. O. P ´ALL-SZAB ´O

Proof. By using the properties of operatorLⁿ, we obtain

L^nC1f .´/D.1 /D^nC1f .´/CI^nC1f .´/ ; ´2U: (2.8) Then (2.7) becomes

.1 /D^nC1f .´/CI^nC1f .´/0

Ch

I^{n 1}f .´/0

I^nC1f .´/0i

h.´/; ´2U:

(2.9) After computation we get

.1 /

D^nC1f .´/0

C

I^{n 1}f .´/0

h.´/

or equivalently

.1 /h

´ Dⁿf .´/₀i₀ Ch

´ Iⁿf .´/₀i₀

h.´/:

The above relation is equivalent to .1 /h

Dⁿf .´/0

C´ Dⁿf .´/00i Ch

Iⁿf .´/0

C´ Iⁿf .´/00i h.´/

or

Lⁿf .´/₀ C´

Lⁿf .´/₀₀

h.´/; ´2U: (2.10) Let

p.´/D.1 /

Dⁿf .´/₀ C

Iⁿf .´/₀ D

Lⁿf .´/₀

; ´2U (2.11) D.1 /

"

´C

1

X

kD2

kⁿa_k´^k

#₀ C

"

´C

1

X

kD2

1 kⁿa_k´^k

#₀ D

D.1 /

"

1C

1

X

kD2

k^nC1a_k´^{k 1}

# C

"

1C

1

X

kD2

1

k^{n 1}a_k´^{k 1}

# D

D1C

1

X

kD2

k^nC1.1 /C 1 k^{n 1}

a_k´^{k 1}D1Cp1´Cp2´²C In view of (2.11), we deduce thatp2HŒ1; 1. Using the notation in (2.11), the (2.10) differential subordination becomes

p.´/C´p⁰.´/h.´/Dq.´/C´q⁰.´/; ´2U:

By using Lemma2we have

p.´/q.´/ i:e:

Lⁿf .´/0

q.´/; ´2U:

Remark5. IfD0we get Theorem 2 from Oros [6] and forD1we get Theorem 2 from Bˇalˇaet¸i [1].

Example3. ForD0,nD1,f 2Awe deduce that

f⁰.´/C3´f⁰⁰.´/C´²f⁰⁰⁰.´/1C2´; ´2U implies

f⁰.´/C´f⁰⁰.´/1C´; ´2U:

Theorem 4. Leth2H.U /such thath.0/D1and

<

1C´h⁰⁰.´/

h⁰.´/

> 1

2; ´2U:

Iff 2Asatisfies the differential subordination L^nC1f .´/0

Ch

I^{n 1}f .´/0

I^nC1f .´/0i

h.´/; ´2U (2.12) then

Lⁿf .´/₀

q.´/; ´2U where q is given byq.´/D 1

´ Z ´

0

h.t /dt. The function q is convex and is the best dominant.

Proof. If we use the differential subordination technique we can see that the func- tiongis convex.[3], p. 66 By using (2.11) we obtain

L^nC1f .´/0

Ch

I^{n 1}f .´/0

I^nC1f .´/0i

Dp.´/C´p⁰.´/; ´2U Then (2.12) becomes

p.´/C´p⁰.´/h.´/; ´2U:

Sincep2HŒ1; 1, we deduce thatp.´/q.´/, i.e.

Lⁿf .´/0

q.´/D 1

´ Z ´

0

h.t /dt; ´2U

andqis the best dominant.

Remark6. IfD0we get Theorem 3 from Oros [6].

Example4. ForD0,nD0,h.´/D1C´

1 ´ we deduce that f⁰.´/C´f⁰⁰.´/ 1C´

1 ´; ´2U;

implies

f⁰.´/1 2

´ln .1 ´/ ; ´2U:

1102 A. O. P ´ALL-SZAB ´O

Theorem 5. Leth2H.U /such thath.0/D1and

<

1C´h⁰⁰.´/

h⁰.´/

> 1

2; ´2U:

Iff 2Asatisfies the differential subordination Lⁿf .´/₀

h.´/; ´2U (2.13)

then Lⁿf .´/

´ q.´/; ´2U where q is given byq.´/D 1

´ Z ´

0

h.t /dt. The function q is convex and is the best dominant.

Proof. If we use the differential subordination technique we can see that the func- tiongis convex. [3], p. 66. Differentiating both sides in (2.2) we obtain

Lⁿf .´/₀

Dp.´/C´p⁰.´/; ´2U Then (2.13) becomes

p.´/C´p⁰.´/h.´/; ´2U:

Sincep2HŒ1; 1, we deduce thatp.´/q.´/, i.e.

Lⁿf .´/

´ q.´/D1

´ Z ´

0

h.t /dt; ´2U

andqis the best dominant.

Remark7. IfD0we get Theorem 5 from Oros [6] and forD1we get Theorem 5 from Bˇalˇaet¸i [1].

Example5. ForD0,nD1,h.´/D 1

.1C´/² we deduce that f⁰.´/ 1

.1C´/²; ´2U;

implies

f .´/

´ 1

1C´; ´2U:

We get the same result as [4].

Definition 5([7], [9], [1], [6]). If0ˇ < 1andn2N, we letL^m_n.ˇ/stand for the class of functionsf 2Am, which satisfy the inequality

<

Lⁿf .´/₀

> ˇ; .´2U / : Remark8. FornD0we obtain<f⁰.´/ > ˇ.

Theorem 6. The setL^m_n.ˇ/is convex.

Proof. Let the function

fi.´/D´C

1

X

kD2

a_ki´^k; iD1; 2 ´2U be in the classL^m_n .ˇ/. It is sufficient to show that the function

h.´/D1f1.´/C2f2.´/

with1; 20and1C2D1is inLn.ˇ/. Since h.´/D´C

1

X

kD2

1a_k1C2a_k2

´^k; ´2U then

Lⁿh .´/D´C

1

X

kD2

kⁿ.1 /C 1 kⁿ

1a_k1C2a_k2

´^k; ´2U: (2.14) Differentiating (2.14), we get

Lⁿh .´/0

D1C

1

X

kD2

k^nC1.1 /C 1 k^{n 1}

1a_k1C2a_k2

´^{k 1}: Hence

<

Lⁿh .´/0

D1C <

( 1

1

X

kD2

k^nC1.1 /C 1 k^{n 1}

a_k1´^{k 1} )

C

C <

( 2

1

X

kD2

k^nC1.1 /C 1 k^{n 1}

ak₂´^{k 1} )

: (2.15)

Sincef1; f22L^m_n.ˇ/, we obtain

<

( i

1

X

kD2

k^nC1.1 /C 1 k^{n 1}

a_ki´^{k 1} )

> i.ˇ 1/ ; iD1; 2: (2.16) Using (2.16) we get from (2.15)

<

Lⁿh .´/₀

> 1C1.ˇ 1/C2.ˇ 1/ ; and since1C2D1, we deduce

<

Lⁿh .´/₀

> ˇ; .´2U /

i.e.L^m_n.ˇ/is convex.

1104 A. O. P ´ALL-SZAB ´O

Theorem 7. If0ˇ < 1andm; n2Nthen we have L^m_n .ˇ/L^m_nC1.ı/ ; whereı .ˇ; m/D2ˇ 1C2 .1 ˇ/ 1

m 1

m

and .x/D Z ´

0

t^{x 1}

1Ctdt. The result is sharp.

Proof. Assume thatf 2L^m_n.ˇ/. LetLⁿf .´/D´p.´/; ´2U. Differentiating, we obtain

Lⁿf .´/₀

Dp.´/C´p⁰.´/; ´2U:

Sincef 2L^m_n.ˇ/, from Definition5we have

< p.´/C´p⁰.´/

> ˇ; ´2U which is equivalent to

p.´/C´p⁰.´/ 1C.2ˇ 1/ ´

1C´ h.´/; ´2U By using Lemma1, we have:

p.´/q.´/h.´/; ´2U;

where

q.´/D 1 m´^m1

Z ´ 0

1C.2ˇ 1/ t

1Ct t^{m1 1}dtD D 1

m´^m1 Z ´

0

2ˇ 1C2.1 ˇ/ 1 1Ct

t^{m1 1}dtD

D 1 m´^m1

Z ´ 0

.2ˇ 1/ t^{m1 1}dtC2 .1 ˇ/

m´^m1 Z ´

0

t^{m1 1} 1Ct dtD D2ˇ 1C2 .1 ˇ/ 1

m 1

m 1

´^m1

; ´2U:

The functionqis convex and is the best dominant. Fromp.´/q.´/follows that

<p.´/ ><q.1/Dı .ˇ; m/D2ˇ 1C2 .1 ˇ/ 1 m

1 m

;

from which we deduce thatL^m_n .ˇ/L^m_nC1.ı/.

Remark9. IfD0we get Theorem 1 from Oros [6] and forD1we get Theorem 1 from Bˇalˇaet¸i [1].

Theorem 8. Letqbe a convex function inU withq.0/D1and let h.´/Dq.´/C 1

cC2´q⁰.´/; ´2U;

wherecis a complex number, with<c > 2.

Iff 2L^m_n.ˇ/andF DIc.f /, where F .´/DIc.f / .´/DcC2

´^cC1 Z ´

0

t^cf .t /dt; <c > 2; (2.17) then

Lⁿf .´/0

h.´/; ´2U; (2.18)

implies

LⁿF .´/₀

q.´/; ´2U;

and this result is sharp.

Proof. From (2.17), we have

´^cC1F .´/D.cC2/

Z ´ 0

t^cf .t /dt; <c > 2; ´2U: (2.19) Differentiating, with respect to z, we obtain

.cC1/F .´/C´F⁰.´/D.cC2/f .´/; ´2U and

.cC1/LⁿF .´/C´

LⁿF .´/0

D.cC2/Lⁿf .´/; ´2U: (2.20) Differentiating (2.20), we obtain

LⁿF .´/₀ C ´

cC2

LⁿF .´/₀₀ D

Lⁿf .´/₀

; ´2U: (2.21) Using (2.21), the differential subordination (2.18) becomes

LⁿF .´/0

C 1 cC2´

LⁿF .´/00

h.´/Dq.´/C 1

cC2´q⁰.´/; ´2U: (2.22) Let

p.´/D

LⁿF .´/₀ D

(

´C

1

X

kD2

kⁿ.1 /C 1 kⁿ

ak´^k

)0

D (2.23)

D1Cp₁´Cp₂´²C ; ´2U; p2HŒ1; 1 : Replacing (2.23) in (2.22) we obtain

p.´/C 1

cC2´p⁰.´/h.´/Dq.´/C 1

cC2´q⁰.´/; ´2U;

Using Lemma1, we obtainp.´/q.´/i.e.

LⁿF .´/₀

q.´/; ´2U;

and q is the best dominant.

1106 A. O. P ´ALL-SZAB ´O

Remark10. IfD0we get Theorem 2.2 from Tˇaut et alii [9].

Example6. If we takecD1C2i andq.´/D1C´ 1 ´ then h.´/D 1 ´²

.3C2i /C2´

.3C2i / .1 ´/² . From Theorem8we deduce

Lⁿf .´/0

1 ´²

.3C2i /C2´

.3C2i / .1 ´/² ; ´2U;

implies

LⁿF .´/0

1C´

1 ´; ´2U;

whereF is given by (2.17).

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[6] G. Oros , “On a class of holomorphic functions defined by Sˇalˇagean differential operator,”Complex Variables., vol. 50, no. 4, pp. 257–264, 2005.

[7] G. Oros and G. I. Oros , “A Class of Holomorphic Function II,” Libertas Math., vol. 23, pp. 65–68, 2003.

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1013, pp. 362–372, 1983.

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Author’s address

A. O. P´all-Szab´o´

Babes¸-Bolyai University, Cluj-Napoca, Romania E-mail address:pallszaboagnes@math.ubbcluj.ro