AfafA.AliAbubakerandMaslinaDarusdownloads.hindawi.com/journals/ijmms/2014/628972.pdf · Research Article On Harmonic Functions Defined by Differential Operator with Respect to -Symmetric
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Research ArticleOn Harmonic Functions Defined by Differential Operator withRespect to 119896-Symmetric Points
Afaf A Ali Abubaker and Maslina Darus
School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 BangiSelangor D Ehsan Malaysia
Correspondence should be addressed to Maslina Darus maslinaukmedumy
Received 29 April 2014 Revised 4 July 2014 Accepted 7 July 2014 Published 23 July 2014
Academic Editor Heinrich Begehr
Copyright copy 2014 A A A Abubaker and M Darus This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We introduce new classes 119872119867120590119904
119896(120582 120575 120572) and 119872119867
120590119904
119896(120582 120575 120572) of harmonic univalent functions with respect to 119896-symmetric points
defined by differential operator We determine a sufficient coefficient condition representation theorem and distortion theorem
1 Introduction
A continuous function 119891 = 119906 + 119894V is a complex valuedharmonic function in a complex domain119862 if both 119906 and V arereal harmonic in 119862 In any simply connected domain 119863 sub 119862
we can write 119891(119911) = ℎ + 119892 where ℎ and 119892 are analytic in119863 We call ℎ the analytic part and 119892 the coanalytic part of119891 A necessary and sufficient condition for 119891 to be locallyunivalent and sense preserving in 119863 is that |ℎ1015840(119911)| gt |119892
1015840
(119911)|
in 119863 See Clunie and Shell-Small (see [1])Thus for 119891 = ℎ + 119892 isin 119878
lowast
119867 we may write
ℎ (119911) = 119911 +
infin
sum
119899=2
119886119899119911119899
119892 (119911) =
infin
sum
119899=1
119887119899119911119899
10038161003816100381610038161198871
1003816100381610038161003816 lt 1 (1)
Note that 119878lowast119867 reduces to 119878lowast the class of normalized analytic
univalent functions if the coanalytic part of 119891 = ℎ + 119892 isidentically zero Also denote by 119878119867 the subclasses of 119878
lowast
119867
consisting of functions 119891 that map 119880 onto starlike domainA function 119891 is said to be starlike of order 120572 in119880 denoted
119891 (119903119890119894120579) minus 119891 (minus119903119890119894120579) ge 120572 (4)
In [8] the authors introduced and studied the class 119878119867119896(120572)
which denotes the class of complex-valued sense-preservingharmonic univalent functions 119891 of the form (1) and
ℎ119896(119911) = 119911 +
infin
sum
119899=2
120601119899119886119899119911119899
119892119896(119911) =
infin
sum
119899=1
120601119899119887119899119911119899
10038161003816100381610038161198871
1003816100381610038161003816 lt 1
(5)
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 628972 6 pageshttpdxdoiorg1011552014628972
2 International Journal of Mathematics and Mathematical Sciences
where
120601119899=
1
119896
119896minus1
sum
]=0120576(119899minus1)]
(119896 ge 1 120576119896
= 1) (6)
From the definition of 120601119899we know
120601119899=
1 119899 = 120580119896 + 1
0 119899 = 120580119896 + 1(119899 ge 2 120580 119896 ge 1) (7)
The differential operator 119863120590119904
120582120575was introduced by Ali Abu-
baker and Darus [9] We define the differential operator ofthe harmonic function 119891 = ℎ + 119892 given by (5) as
119863120590119904
120582120575119891 (119911) = 119863
120590119904
120582120575ℎ (119911) + (minus1)
119904
119863120590119904
120582120575119892 (119911) (8)
where
119863120590119904
120582120575ℎ (119911) = 119911 +
infin
sum
119899=2
120595119899(120582 120575 120590 119904) 119886
119899119911119899
119863120590119904
120582120575119892 (119911) =
infin
sum
119899=1
120595119899(120582 120575 120590 119904) 119887
119899119911119899
(9)
and also 120595119899(120582 120575 120590 119904) = 119899
119904
(119862(120575 119899)[1 + 120582(119899 minus 1)])120590 120582 ge 0
119862(120575 119899) = (120575 + 1)119899minus1
(119899 minus 1) for 120575 120590 119904 isin 1198730= 0 1 2
and (119909)119899is the Pochhammer symbol defined by
We note that when 119904 = 0 120590 = 1 and 120582 = 0we obtain the Rus-cheweyh derivative for harmonic functions (see [7]) when120590 = 0weobtain the Salagean operator for harmonic functions(see [10]) and when 120590 = 1 119904 = 0 we obtain the operator forharmonic functions given by Al-Shaqsi and Darus [11]
Let 119872119867120590119904
119896(120582 120575 120572) denote the class of complex-valued
sense-preserving harmonic univalent functions 119891 of theform (5) which satisfy the condition
The above-required condition must hold for all values of 119911|119911| = 119903 lt 1 Upon choosing the values of 119911 on the positivereal axis where 0 le 119911 = 119903 lt 1 we must have
((1 minus 120572) minus
infin
sum
119899=2
120595119899(120582 120575 120590 119904) (119899 minus 120572120601
If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis
contradicts the required condition for 119891 isin 119872119867120590119904
119896(120582 120575 120572)
and the proof is complete
Now the distortion result is given
Theorem 5 If 119891 isin 119872119867120590119904
119896(120582 120575 120572) then
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) minus
Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867
120590119904
119896(120582 120575 120572) Taking
the absolute value of 119891(119911) we obtain
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) 119903 minus
Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors read and approved the final paper
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984
[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990
[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959
[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006
[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979
[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977
[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004
[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007
[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011
6 International Journal of Mathematics and Mathematical Sciences
[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002
[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009
[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012
[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009
[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009
[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009
We note that when 119904 = 0 120590 = 1 and 120582 = 0we obtain the Rus-cheweyh derivative for harmonic functions (see [7]) when120590 = 0weobtain the Salagean operator for harmonic functions(see [10]) and when 120590 = 1 119904 = 0 we obtain the operator forharmonic functions given by Al-Shaqsi and Darus [11]
Let 119872119867120590119904
119896(120582 120575 120572) denote the class of complex-valued
sense-preserving harmonic univalent functions 119891 of theform (5) which satisfy the condition
The above-required condition must hold for all values of 119911|119911| = 119903 lt 1 Upon choosing the values of 119911 on the positivereal axis where 0 le 119911 = 119903 lt 1 we must have
((1 minus 120572) minus
infin
sum
119899=2
120595119899(120582 120575 120590 119904) (119899 minus 120572120601
If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis
contradicts the required condition for 119891 isin 119872119867120590119904
119896(120582 120575 120572)
and the proof is complete
Now the distortion result is given
Theorem 5 If 119891 isin 119872119867120590119904
119896(120582 120575 120572) then
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) minus
Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867
120590119904
119896(120582 120575 120572) Taking
the absolute value of 119891(119911) we obtain
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) 119903 minus
Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors read and approved the final paper
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984
[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990
[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959
[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006
[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979
[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977
[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004
[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007
[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011
6 International Journal of Mathematics and Mathematical Sciences
[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002
[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009
[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012
[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009
[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009
[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009
The above-required condition must hold for all values of 119911|119911| = 119903 lt 1 Upon choosing the values of 119911 on the positivereal axis where 0 le 119911 = 119903 lt 1 we must have
((1 minus 120572) minus
infin
sum
119899=2
120595119899(120582 120575 120590 119904) (119899 minus 120572120601
If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis
contradicts the required condition for 119891 isin 119872119867120590119904
119896(120582 120575 120572)
and the proof is complete
Now the distortion result is given
Theorem 5 If 119891 isin 119872119867120590119904
119896(120582 120575 120572) then
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) minus
Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867
120590119904
119896(120582 120575 120572) Taking
the absolute value of 119891(119911) we obtain
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) 119903 minus
Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors read and approved the final paper
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984
[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990
[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959
[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006
[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979
[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977
[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004
[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007
[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011
6 International Journal of Mathematics and Mathematical Sciences
[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002
[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009
[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012
[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009
[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009
[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009
The above-required condition must hold for all values of 119911|119911| = 119903 lt 1 Upon choosing the values of 119911 on the positivereal axis where 0 le 119911 = 119903 lt 1 we must have
((1 minus 120572) minus
infin
sum
119899=2
120595119899(120582 120575 120590 119904) (119899 minus 120572120601
If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis
contradicts the required condition for 119891 isin 119872119867120590119904
119896(120582 120575 120572)
and the proof is complete
Now the distortion result is given
Theorem 5 If 119891 isin 119872119867120590119904
119896(120582 120575 120572) then
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) minus
Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867
120590119904
119896(120582 120575 120572) Taking
the absolute value of 119891(119911) we obtain
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) 119903 minus
Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors read and approved the final paper
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984
[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990
[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959
[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006
[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979
[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977
[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004
[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007
[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011
6 International Journal of Mathematics and Mathematical Sciences
[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002
[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009
[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012
[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009
[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009
[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009
If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis
contradicts the required condition for 119891 isin 119872119867120590119904
119896(120582 120575 120572)
and the proof is complete
Now the distortion result is given
Theorem 5 If 119891 isin 119872119867120590119904
119896(120582 120575 120572) then
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) minus
Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867
120590119904
119896(120582 120575 120572) Taking
the absolute value of 119891(119911) we obtain
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus
100381610038161003816100381611988711003816100381610038161003816) 119903 minus
Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors read and approved the final paper
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984
[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990
[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959
[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006
[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979
[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977
[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004
[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007
[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011
6 International Journal of Mathematics and Mathematical Sciences
[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002
[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009
[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012
[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009
[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009
[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009
6 International Journal of Mathematics and Mathematical Sciences
[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002
[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009
[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012
[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009
[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009
[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009