Research Article Certain Subclasses of Analytic Functions with Complex … · 2019. 7. 31. · Certain Subclasses of Analytic Functions with Complex Order A.Selvam, 1 P.SooriyaKala,
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Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 958796 6 pageshttpdxdoiorg1011552013958796
Research ArticleCertain Subclasses of Analytic Functions with Complex Order
A Selvam1 P Sooriya Kala1 and N Marikkannan2
1 Department of Mathematics VHNSN College Virudhunagar 626001 India2Department of Mathematics Government Arts College Melur 625106 India
Correspondence should be addressed to N Marikkannan natarajanmarikkannangmailcom
Received 12 August 2013 Accepted 12 September 2013
Academic Editors G Bonanno and A Ibeas
Copyright copy 2013 A Selvam et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Two new subclasses of analytic functions of complex order are introduced Apart from establishing coefficient bounds for theseclasses we establish inclusion relationships involving (119899-120575) neighborhoods of analytic functionswith negative coefficients belongingto these subclasses
1 Introduction
LetA(119899) denote the class of functions of the form
119891 (119911) = 119911 minus
infin
sum
119896=119899+1
119886119896119911119896
(119886119896ge 0 119899 isin N = 1 2 3 ) (1)
which are analytic and univalent in the open disc
Δ = 119911 isin C |119911| lt 1 (2)
A function 119891(119911) isin A(119899) is star-like of complex order 119887denoted as 119891(119911) isin 119878
lowast
(119887) if and only if it satisfies
R1 +
1
119887
(
1199111198911015840
119891
minus 1) gt 0 (119911 isin Δ) (3)
A function 119891(119911) isin A(119899) is convex of complex order 119887denoted as 119891(119911) isin 119862(119887) if and only if it satisfies
R1 +
1
119887
(
11991111989110158401015840
1198911015840) gt 0 (119911 isin Δ) (4)
These classes 119878lowast(119887) and 119862(119887) are introduced and studied byNasr and Aouf [1] and Wiatrowski [2]
For the two functions 119891119895(119895 = 1 2) given by
119891119895(119911) = 119911 +
infin
sum
119896=2
119886119896119895
119911119896
(119895 = 1 2) (5)
the Hadamard product or convolution denoted by (1198911
lowast
1198912)(119911) is given by
(1198911lowast 1198912) (119911) = 119911 +
infin
sum
119896=2
1198861198961
1198861198962
119911119896
(6)
Given 119891(119911) of the form (1) and 120575 ge 0 we define 119899-120575neighborhood of a function 119891 isin A(119899) as
For suitable values of 1205721198941015840119904 1205731198951015840119904 119902 119904 119898 120582 and 120583 we can
deduce several operators such as Salagean derivative operator[6] Ruscheweyh derivative operator [7] fractional calculusoperator [8] Carlson-Shaffer operator [9] Dziok-Srivatsavaoperator [10] and also the operator introduced by Abubakerand Darus [11]
Definition 1 For 0 le 120572 le 1 we let 119860 be the subclass ofA(119899)
consisting of functions of the form (1) that satisfy
where 119911 isin Δ 119887 isin C 0 0 lt 120574 le 1 andD119898120582120583
(1205721 1205731)119891(119911) are
as given in (15)
By specializing the parameters involved in the abovedefinitions we could arrive at several known as well as newclasses For example by taking 120582 = 1 120583 = 0 119902 = 2 119904 = 11205721= 1205731 and 120572
2= 1 and the above classes reduced to
1198601= 119891 isin A (119899) |
1003816100381610038161003816100381610038161003816
1
119887
(119911[D119898
119891 (119911)]1015840
times ( (1 minus 120572)D119898
119891 (119911)
+ 120572119911[D119898
119891 (119911)]1015840
)
minus1
)
1003816100381610038161003816100381610038161003816
lt 120574
(18)
The Scientific World Journal 3
where D119898119891(119911) denote the Salagean derivative of order 119898
Further by taking 119898 = 0 in the definition of the classes119860 and 119861 we could arrive at 119878
119899(119902 119904 120572 119887 120574) and 119877
119899(119902 119904 120572 119887 120574)
whichwere introduced and studied byMurugusundaramoor-thy et al [12]
In this paper we establish the coefficient inequalitiesfor the classes 119860 and 119861 and several inclusion relationshipsinvolving 119899-120575 neighborhoods of analytic univalent functionswith negative and missing coefficients belonging to theseclasses
2 Coefficient Inequalities
Theorem 3 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860 if and only ifinfin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
For suitable values of 1205721198941015840119904 1205731198951015840119904 119902 119904 119898 120582 and 120583 we can
deduce several operators such as Salagean derivative operator[6] Ruscheweyh derivative operator [7] fractional calculusoperator [8] Carlson-Shaffer operator [9] Dziok-Srivatsavaoperator [10] and also the operator introduced by Abubakerand Darus [11]
Definition 1 For 0 le 120572 le 1 we let 119860 be the subclass ofA(119899)
consisting of functions of the form (1) that satisfy
where 119911 isin Δ 119887 isin C 0 0 lt 120574 le 1 andD119898120582120583
(1205721 1205731)119891(119911) are
as given in (15)
By specializing the parameters involved in the abovedefinitions we could arrive at several known as well as newclasses For example by taking 120582 = 1 120583 = 0 119902 = 2 119904 = 11205721= 1205731 and 120572
2= 1 and the above classes reduced to
1198601= 119891 isin A (119899) |
1003816100381610038161003816100381610038161003816
1
119887
(119911[D119898
119891 (119911)]1015840
times ( (1 minus 120572)D119898
119891 (119911)
+ 120572119911[D119898
119891 (119911)]1015840
)
minus1
)
1003816100381610038161003816100381610038161003816
lt 120574
(18)
The Scientific World Journal 3
where D119898119891(119911) denote the Salagean derivative of order 119898
Further by taking 119898 = 0 in the definition of the classes119860 and 119861 we could arrive at 119878
119899(119902 119904 120572 119887 120574) and 119877
119899(119902 119904 120572 119887 120574)
whichwere introduced and studied byMurugusundaramoor-thy et al [12]
In this paper we establish the coefficient inequalitiesfor the classes 119860 and 119861 and several inclusion relationshipsinvolving 119899-120575 neighborhoods of analytic univalent functionswith negative and missing coefficients belonging to theseclasses
2 Coefficient Inequalities
Theorem 3 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860 if and only ifinfin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
Further by taking 119898 = 0 in the definition of the classes119860 and 119861 we could arrive at 119878
119899(119902 119904 120572 119887 120574) and 119877
119899(119902 119904 120572 119887 120574)
whichwere introduced and studied byMurugusundaramoor-thy et al [12]
In this paper we establish the coefficient inequalitiesfor the classes 119860 and 119861 and several inclusion relationshipsinvolving 119899-120575 neighborhoods of analytic univalent functionswith negative and missing coefficients belonging to theseclasses
2 Coefficient Inequalities
Theorem 3 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860 if and only ifinfin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007