A linear operator of a new class of multivalent harmonic functions

Transcription

1 International Journal of Scientific Research Publications, Volume 7, Issue 8, August A linear operator of a new class of multivalent harmonic functions * WaggasGalibAtshan, ** Ali Hussein Battor, *** Amal Mohammed Darweesh * Department of Mathematics, College of Computer Science Mathematics, University of Al-Qadisiya, Diwaniya-Iraq **,*** Department of Mathematics, College of Education for Girls, University of Kufa, Najaf Iraq Abstract- New classes of multivalently functions with a linear operator are introduced. We give sufficient coefficient bounds for ff(zz) KKMM pp (KK, ββ, qq) then we show that these sufficient coefficient conditions are also necessary for ff(zz) AAMM pp (KK, ββ, qq). Furthermore, we determine extreme points, convex combination, convolution property integral operator for these functions. Also we obtain new results in this paper. Index Terms- Multivalent harmonic functions, Coefficient bounds, Extreme points, Convex combination, Integral operator. AMS subject classification: 30C45. A I. INTRODUCTION continuous function ff = uu + iiii is a complex valued harmonic function in a complex domain Cif both uu aaaaaa vv are real harmonic in C. In any simply connected domain DD C,we can write ff = h + gg, where h aaaaaa gg are analytic in DD. We call h the analytic part gg the co-analytic part of ff. A necessary sufficient condition for ff to be locally univalent sensepreserving in DD is that h (zz) > gg (zz) in DD, see Clunie Sheil-Small [5]. Denote by MM(pp) the class of functions ff = h + gg that are harmonic multivalent sense-preserving in the unit disk UU = {zz C: zz < 1}. The class MM(pp) was studied by Ahuja Jahangiri [1] class MM(pp) for pp = 1 was defined studied by Jahangiri et. al. in [6]. For ff = h + gg MM(pp), we may express the analytic functions h aaaaaa gg as: h(zz) = zz pp + aa nn zz nn, gg(zz) = bb nn zz nn Let WW pp denote the subclass of MM(pp) consisting of functions ff = h + gg, where h aaaaaa gg are given by: h(zz) = zz pp aa nn zz nn, gg(zz) = bb nn zz nn Now, we define anew class KKMM pp (KK, ββ, qq) of harmonic functions of the form (1) that satisfy the inequality where 0 ββ < 1 pp, pp > qq, pp N = {1,2, }, qq N 0 = N {0}, 0 KK 1, λλ 0, γγ 0, δδ(ii, jj) =, bb pp < 1. (1), bb pp < 1. (2) KK DD pp (λλ, ββ, γγ)ff(zz) qq + zz 2 DD pp (λλ, ββ, γγ)ff(zz) qq+2 RRRR zz DD pp (λλ, ββ, γγ)ff(zz) qq+1 > ββ, (3) ff qq (zz) = zz ppqq + δδ(nn, qq)aa nn zz nnqq, nn=1 ii! jj = 0 = 1 (ii jj)! ii(ii 1) (ii jj + 1) jj 0, DD pp (λλ, ββ, γγ)ff(zz) = DD pp (λλ, ββ, γγ)h(zz) + DD pp (λλ, ββ, γγ)gg(zz). (4) The operator DD pp (λλ, ββ, γγ) denotes the linear operator introduced in [8]. For h aaaaaa gg given by (1), we obtain (nn DD pp (λλ, ββ, γγ)h(zz) = zz pp pp)λλ aa nn zz nn, (5) DD pp (λλ, ββ, γγ)gg(zz) = 1 + bb nn zz nn, (6)

2 International Journal of Scientific Research Publications, Volume 7, Issue 8, August where pp N = {1,2, }, λλ 0, γγ 0, 0. We further denote by AAMM pp (KK, ββ, γγ) the subclass of KKMM pp (KK, ββ, γγ) that satisfies the relation AAMM pp (KK, ββ, γγ) = AAMM pp KKMM pp (KK, ββ, γγ). (7) II. COEFFICIENT INEQUALITY We need the following lemma in our results: Lemma 1[2]:Re{ww(zz)} > ββ iiii aaaaaa oooooooo iiii ww(zz) (1 + ββ) ww(zz) + (1 ββ). In the following theorem, we find a coefficient inequality for functions in the class KKMM pp (KK, ββ, γγ). Theorem 1: Let ff = h + gg (h aaaaaa gg being given by (1)). If aa nn δδ(nn, qq) (nn qq) (nn qq 1)KK ββ + KK 1 + bb nn, (8) where 0 ββ < 1 pp, pp > qq, pp N = {1,2, }, qq N 0 = N {0}, 0 KK 1, 0, λλ 0, γγ 0, then ff is harmonic p-valent sensepreserving in UU ff KKMM pp (KK, ββ, γγ). Proof : Let KK DD pp (λλ, ββ, γγ)ff(zz) qq + zz 2 DD pp (λλ, ββ, γγ)ff(zz) qq+2 ww(zz) = zz DD pp (λλ, ββ, γγ)ff(zz) qq+1. Using the fact in Lemma (1) Re{ww(zz)} > ββ iiii aaaaaa oooooooo iiii ww(zz) (1 + ββ) ww(zz) + (1 ββ). (9) Substituting for ww resorting to simple calculations, we find that ww(zz) (1 + ββ) (pp qq) (pp qq 1)KK (1 + ββ) + KK zz ppqq + δδ(nn, qq) (nn qq) (nn qq 1)KK (1 + ββ) + KK aa nn zz nnqq + δδ(nn, qq) (nn qq) (nn qq 1)KK (1 + ββ) + KK bb nn zz nnqq, (10) ww(zz) + (1 ββ) (pp qq) (pp qq 1)KK + (1 ββ) + KK zz ppqq δδ(nn, qq) (nn qq) (nn qq 1)KK + (1 ββ) + KK aa nn zz nnqq Evidently (10) (11) in conjunction with (8) yields ww(zz) (1 + ββ) ww(zz) + (1 ββ) 0. The harmonic functions where ff(zz) = zz pp + δδ(nn, qq) (nn qq) (nn qq 1)KK + (1 ββ) + KK bb nn zz nnqq. (11) xx nn xx nn yy nn + + yy nn =, show that the coefficients bounds given by (8) is sharp. The functions of the form (12) are in KKMM pp (KK, ββ, γγ) because in view of (12) infer that zz nn zz, nn (12)

3 International Journal of Scientific Research Publications, Volume 7, Issue 8, August = xx nn + yy nn =. aa nn bb nn The restriction placed in Theorem (1) on the moduli of coefficients of ff = h + gg implies for arbitrary rotation of the coefficients of ff, the resulting functions would still be harmonic multivalent ff KKMM pp (KK, ββ, γγ). The following theorem shows that the condition (8) is also necessary for function ff to belong to AAMM pp (KK, ββ, γγ). Theorem 2: Let ff = h + gg with h aaaaaa gg are given by (2). Then ff AAMM pp (KK, ββ, γγ) if only if aa nn + where 0 ββ < 1 pp, pp > qq, pp N = {1,2, }, qq N 0 = N {0}, 0 KK 1, 0, λλ 0, γγ 0. bb nn, (13) Proof : By noting that AAMM pp (KK, ββ, γγ) KKMM pp (KK, ββ, γγ), the sufficiency part of Theorem (2) follows at once from Theorem (1). To prove the necessary part, let us assume thatff AAMM pp (KK, ββ, γγ). Using (3), we get KK DDpp (λλ, ββ, γγ)h(zz) qq qq + DD pp (λλ, ββ, γγ)gg(zz) + zz 2 DD pp (λλ, ββ, γγ)h(zz) qq+2 qq+2 + DD pp (λλ, ββ, γγ)gg(zz) RRRR zz DD pp (λλ, ββ, γγ)h(zz) qq+1 qq+1 + DD pp (λλ, ββ, γγ)gg(zz) KKKK KKcc 1 + (nnpp)λλ aa nn zz nnqq KKKK 1 + (nnpp)λλ bb nn zz nnqq = RRRR δδ(pp, qq + 1) δδ(nn, qq + 1) 1 + (nnpp)λλ aa nn zz nnqq δδ(nn, qq + 1) 1 + (nnpp)λλ bb nn zz nnqq > αα, where tt = [1 + (pp qq)(pp qq 1)] aaaaaa cc = δδ(nn, qq)[1 + (nn qq)(nn qq 1)]. If we choose zz to be real let zz 1, we obtain the condition (13). III. EXTREME POINTS Next, we determine the extreme points of the closed convex hull of AAMM pp (KK, ββ, γγ), denoted by AAMM pp (KK, ββ, γγ). Theorem 3:ff AAMM if pp (KK, ββ, γγ) only if wherezz UU, h pp (zz) = zz pp, (nn = pp + 1, pp + 2, ) (nn = pp, pp + 1, ) (μμ nn + θθ nn ) = 1, (μμ nn 0, θθ nn 0). In particular, the extreme points of AAMM pp (KK, ββ, γγ) are {h nn } aaaaaa {gg nn }. ff(zz) = (μμ nn h nn + θθ nn gg nn ), (14) h nn (zz) = zz pp zz nn, (15) gg nn (zz) = zz pp zz nn, (16)

4 International Journal of Scientific Research Publications, Volume 7, Issue 8, August Proof : Suppose ff is of the from (14). Using (15) (16), we get Then = (μμ nn + θθ nn ) zz pp ff(zz) = (μμ nn h nn + θθ nn gg nn ) μμ nn zz nn = zz pp which implies that ff AAMM pp (KK, ββ, γγ). Conversely, assume that ff AAMM pp (KK, ββ, γγ). Putting (nn = pp + 1, pp + 2, ), (nn = pp, pp + 1, ), we get θθ nn (zz) nn μμ nn zz nn θθ nn (zz) nn. + μμ nn = (μμ nn + θθ nn ) μμ pp = 1 μμ pp, μμ nn = θθ nn = ff(zz) = (μμ nn h nn + θθ nn gg nn ). Theorem 4: The class AAMM pp (KK, ββ, γγ) is a convex set. Proof : Let the function ff nn,jj (jj = 1,2)defined by ff nn,jj (zz) = zz pp aa nn,jj zz nn bb nn,jj zz nn be in the class AAMM pp (KK, ββ, γγ). It is sufficient to prove that the function HH(zz) = ττff nn,1 (zz) + (1 ττ)ff nn,2 (zz), (0 ττ < 1), is also in the class AAMM pp (KK, ββ, γγ). Since for 0 ττ < 1, with the aid of Theorem (2), we have aa nn, bb nn, HH(zz) = zz pp ττ aa nn,1 + (1 ττ) aa nn,2 zz nn ττ bb nn,1 + (1 ττ) bb nn,2 (zz) nn θθ nn

5 International Journal of Scientific Research Publications, Volume 7, Issue 8, August = ττ + ττ aa nn,1 + (1 ττ) aa nn,2 ττ bb nn,1 + (1 ττ) bb nn,2 bb nn,1 +(1 ττ) + Hence, HH(zz) AAMM pp (KK, ββ, γγ). For harmonic functions We define the convolution of ff aaaaaa FF as ττττ(pp, qq) + (1 ττ) =. bb nn,2 aa nn,1 aa nn,2 ff(zz) = zz pp aa nn zz nn bb nn FF(zz) = zz pp AA nn zz nn BB nn (zz) nn (17) (zz) nn. (18) (ff FF)(zz) = zz pp aa nn AA nn zz nn bb nn BB nn (zz) nn. (19) IV. CONVOLUTION PROPERTY In the following theorem we examine the convolution property of the class AAMM pp (KK, ββ, γγ). Theorem 5: If ff aaaaaa FF are in the class AAMM pp (KK, ββ, γγ), then (ff FF) also in the class AAMM pp (KK, ββ, γγ). Proof : Let ff aaaaaa FF of the from (17) (18) belong to AAMM pp (KK, ββ, γγ). Then the convolution of ff aaaaaa FF is given by (19). Note that AA nn 1aaaaaa BB nn 1, since FF AAMM pp (KK, ββ, γγ). Then we can write aa nn AA nn + + bb nn BB nn aa nn The right h side of the above inequality is bounded by because ff AAMM pp (KK, ββ, γγ). Therefore(ff FF) AAMM pp (KK, ββ, γγ). 5- Integral Operator: Definition 1[7]: The June-Kim-Srivastava integral operator is defined by If JJ σσ h(zz) = (pp + 1)σσ zzγ(σσ) zz bb nn. llllll zz tt σσ KK(tt)dddd, σσ > 0. (20) 0

6 International Journal of Scientific Research Publications, Volume 7, Issue 8, August then also JJ σσ is a linear operator. Remark 1: If ff(zz) = h(zz) + gg(zz), where then h(zz) = zz pp aa nn zz nn, Theorem 6: If ff AAMM pp (KK, ββ, γγ), then JJ σσ ff is also in AAMM pp (KK, ββ, γγ). Proof : By (21) (22), we obtain JJ σσ ff(zz) = JJ σσ zz pp aa nn zz nn bb nn (zz) nn since ff AAMM pp (KK, ββ, γγ), then by Theorem (2), we have We must show JJ σσ h(zz) = zz pp pp + 1 σσ nn + 1 aa nn zz nn, (21) h(zz) = zz pp aa nn zz nn, gg(zz) = bb nn zz nn, bb pp < 1, JJ σσ ff(zz) = JJ σσ h(zz) + JJ σσ gg(zz). (22) = zz pp pp + 1 σσ nn + 1 aa nn zz nn pp + 1 σσ nn + 1 aa nn + + bb nn pp + 1 σσ nn + 1 aa nn pp + 1 σσ nn + 1 bb nn bb nn (zz) nn. (23). (24) But in view of (23) the inequality in (24) holds true if pp+1 nn+1 σσ 1, since σσ > 0 aaaaaa pp nn, therefore (24) holds true this gives the result. Theorem 7: Let ff AAMM pp (KK, ββ, γγ). Then bb DD pp (λλ, ββ, γγ)ff(zz) 1 + bb pp zz pp pp + δδ(pp + 1, qq) (pp qq + 1) (pp qq)kk ββ + KK zz pp+1 bb DD pp (λλ, ββ, γγ)ff(zz) 1 bb pp zz pp pp δδ(pp + 1, qq) (pp qq + 1) (pp qq)kk ββ + KK zz pp+1. Proof : Let ff AAMM pp (KK, ββ, γγ), then we have δδ(pp + 1, qq) (pp qq + 1) (pp qq)kk ββ + KK 1 + ( aa nn + bb nn ) Which implies that ( aa nn + bb nn ) Applying this inequality in the following assertion, we obtain DD pp (λλ, ββ, γγ)ff(zz) = zz pp 1 + ( aa nn + bb nn ) bb pp bb pp. δδ(pp + 1, qq) (pp qq + 1) (pp qq)kk ββ + KK aa nn zz nn 1 + bb nn (zz) nn,

7 International Journal of Scientific Research Publications, Volume 7, Issue 8, August bb pp zz pp bb pp zz pp ( aa nn + bb nn ) zz nn ( aa nn + bb nn ) zz pp+1 bb 1 + bb pp zz pp pp + δδ(pp + 1, qq) (pp qq + 1) (pp qq)kk ββ + KK zz pp+1. Also, on the other h we obtain (nn DD pp (λλ, ββ, γγ)ff(zz) 1 bb pp zz pp pp)λλ 1 + ( aa nn + bb nn ) zz nn 1 bb pp zz pp 1 + ( aa nn + bb nn ) zz pp+1 bb 1 bb pp zz pp pp δδ(pp + 1, qq) (pp qq + 1) (pp qq)kk ββ + KK zz pp+1. REFERENCES [1] O. P. Ahuja J. M. Jahangiri, Multivalent harmonic starlike functions, Ann. Univ. Marie-Curie Sklodowska sect. A, 55(1)(2001). [2] E. S. Aqlan, Some problems connected with geometric function theory, Ph. D. Thesis(2004), Pune University, Pune. [3] W. G. Atshan, On a certain class of multivalent functions defined by Hadama product, Journal of Asian Scientific Research, 3(9)(2013), [4] W. G. Atshan S. R. Kulkarni, New classes of multivalentlyharmonic functions, Int. J. of Math. Analysis, Vol. 2, 2008, no, 3, [5] J. G. Clunie T. Shell-Small, Harmonic univalent functions, Aun. Acad. Aci. Fenn. Ser. A. I. Math, 9(1984), [6] J. M. Jahangiri, G. Murugusundaramoorthy, K. Vijaya, Sặlặgean-typeharmonic univalent functions, South. J. Pure Appl. Math., 2(2002), [7] I. B. Jung, Y. C. Kim H. M. Srivastava, The Hardy space of analytic functions associated with one-parameter families of integral operators, J. Math. Appl. 176(1993), [8] H. Mahzoon S. Latha, Neighborhoods of multivalent functions, Int. J. Math. Anal., 3(30)(2009), AUTHORS First Author WaggasGalibAtshan, Department of Mathematics, College of Computer Science Mathematics, University of Al- Qadisiya, Diwaniya-Iraq, waggashnd@gmail.com ; waggas_hnd@yahoo.com Second Author Ali Hussein Battor, Department of Mathematics, College of Education for Girls, University of Kufa, Najaf Iraq, battor_ali@yahoo.com Third Author Amal Mohammed Darweesh, Department of Mathematics, College of Education for Girls, University of Kufa, Najaf Iraq, amalm.alhareezi@uokufa.edu.iq