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I ntnat. J. Mh. & Math. Sci. Vol. 6 No. 2 (1983) 327-334 327 ON A GENERALIZATION OF CLOSE-TO-CONVEXITY K. INAYAT NOOR Mathematics Department Science College of Education for Girls Sitteen Road, Riyadh, Saudi Arabia (Received July 9, 1979 and in revised form August 30, 1982) ABSTRACT. A class T k of analytic functions in the unit disc is defined in which the A necessary condition for a function f concept of close-to-convexity is generalized. to belong to Tk, raduis of convexity problem and a coefficient result are solved in this paper. KEY WORDS AND PHRASES. Close-to-convex functions, functions of bounded boundary rotation, necessary condon, radius of convexly, generalized Koebe function. 1980 tTHEMATICS SUBJECT CLASSIFICATION CODES. 1 Primary 30A32, Secondary 30A34. INTRODUCT ION This paper is directed to mathematical specialists or non-specialists familiar with multivalent functions [i], and to close-to-convex functions [2]. Let Vk be the class of functions of bounded boundary rotation and K be the class of close-to-convex functions. We generalize the concept of close-to-convexity in the following direction. Definition. Let f with f(z) f’(z) Then 0. feTk, k>_2, Re It is clear that T 2 cz + n Y. n=2 a z n be analytic in E {z:Izl<l} Icl=l and if there exist a function geV such that, for zeE k f’(z) >0. g’(z) (1.I) K. E Using a method by Kaplan [2], we have THEOREM i. Let fgT Then with z k. 0 2 f.-l’(z) iRe I(zf’(z))’ 1 re dO > i0 and 0 k 2 1 < @ 2 (1.2) K.I. NOOK 328 REMARK i. From theorem i, we can interpret some geometric meaning for the class T k- For simplicity, let us suppose that the image domain is bounded by an analytic curve C. At a point on C, the outward drawn normal has an angle arg[ei0f (e i0 ]. (1.2), it follows that the angle of the outward drawn normal turns back at This is a necessary condition for a function f to belong to T k. . Then from most k It will be inter- esting to see if this condition is also sufficient. RE,LARK 2. Goodman [3] defines the class K(B) of functions as follows. Let f with f(z) if for z=re i6 + z and Y. a z n n=2 61 62 < n be analytic in E and f’(z) 0. Then for B>0, feK(6), 02 iRe(zf’fI(z)(z))’] d6 > -8 1 We note that 2. TkCK(). MAIN RESULTS From remark 2 and results given in [3] for the class K(B), we have at once THEOREM 2. (i) Let feT k- Denote by L(r,f) the length fo the image of the circle A(r,f) the area of f(Izl=r). Then for Izl r under f and by 0<_r<l, <_ (a) L(r,f) L(r,Fk) _< A(r,Fk) (b) A(r,f) where F is defined by, for zeE, k Fk(Z) (-- [\ i_-I z + A (k)z n I (2.1) n n=2 and clearly Fk e T k. lanl (ii) < An(k) n 2,3 where A (k) is defined by (2.1). (iii) For z re i6 (l-r) k >__2 This result is sharp for each n > 2. 0 < r < i, 1/2k (+/-+r)k+2 < f’(z) < (--+[-)1/2e (l-r) 1/2k+2 These bounds are sharp, equality being attained for the function F k defined by (2.1). GENERALIZATION OF CLOSE-TO-CONVEXITY 329 We also need the following result. Lemma 1 [4]. Let ggVk. Then there are two starlike functions s and s 2 such that I for zgE .gk4 g’(z) (Sl(Z)/Z) (s2 (z)/z) THEOREM 3. fgT k-1/2 k if and only if (k (z)) f’(z) i (k (z)) k-1/2 kl k2ek From definition i, we have PROOF: f’(z) g’(z)h(z), geV k and Re h(z)>0. Using lemma I, we know that there are two starlike functions s (s(z)/z) g’(z) (s2(z)/z) Thus (s (z)/z) f’(z) (s2(z)/z) and k I Lemma 2. 2 and s 2 kk-1/2 1/2k ((Sl (z)h(z))/z) k_. h (z) ((s 2 (z)h(z))/z) 1/2k 1/2k-1/2 are two suitable selected close-to-convex functions. Let H be analytic and be defined as H(z)g’(z) (zg’(z))’, ggVk and H(z) 7+ hl(Z) h2(z)’ Re ho(Z) > 0, i=1,2, h (0)= 1 i Then 2 [H(z) 2 dO <_ IH’(z) Id0 < 1 + (k2-1)r l_r2- 2 (z re i0 0 and 2i 1 2 l-r 2 0 PROOF: such that zgE, k (k (z) k+ (k (z))k-1/2 where k I By the representation formula due to Paatero [5], we can write K.I. NOOR 330 + ze it d(t), +/- 1 H(z) +/- -ze 2-0 where 27 2, and da(t) Id(t) 0 I + I Let H(z) - Th en c i n ICn] <_ k 0 < c z n n=l n 27 [ -in e td.t, and so for n>_l, 0 I Id(t) < k Thus 2 i 2 IH(z)l 2 E n=0 de 0 Aso ICnl 2 2n r 2 2n (k2_l )r 2 i+ Y. r (l+k < l_r2 n=l 2 i H’ (z) e (l_zeit 2 d(t) Thus 27 i 2 iH,(z)id e THEORF21 4: < ll-rei(e+t) 12 Let feT k. Then for deld(t) < l,r’2 Id(t) < l_r 2 n>_l, 2 < c (k)n where c(k) is a constant and depends only on k. PROOF: Since feTk, f’(z) we have for zgE, geVk g’(z)h(z), and Re h(z)>0 Set F(z) z(zf’ (z))’ where Re h(z) > 0 and H(z)g’ (z) i Thus, for E and n>l; zg’ (z)[H(z)h(z) + zh’ (z)], (zg’ (z))’, 2(z)’ with Re h i(z) > 0, i=1,2, h i(0)=l (2,2) GENERALIZATION OF CLOSE-TO-CONVEXITY 331 2 (n+l) 2 an+l-n I-1 anl< 2rn+l and by using lemma i and (2.2), we obtain k- 2g {n+t) 2 an+l-n 2 anl< f iz_ 0 where Sl, s is i (z) IH(z)h(z) Is2(z) lk-1/2 + zh’(z) Id0 (2.3) are starlike functions. 2 It is well-known [i] that for starlike function seS, (l+r) (l-r) (2.4) 2 Then by a result of Golusin [6,p162], there exists a z I with Let 0<r<l. Zll 2 z[ r such that for all z, IZ-Zl] ISl(Z) r, 2r < l_r 2 (2.5) 2 From (2.3)-(2.5), we have hk-1/2 (n+l) 2 an+l-n2 an I<-- n+ll (i<k-1/2(i!r’22)( )r (2 2r NOw as in [7], we have with z re lh(z)12d0 l+3r 2 l_r < 0 I ]H(z)h(z)+zh (z) Id0 (2.6) i0 .2 1 2n 2 2 (2.7) and 2i 2 flzh’(z) Id8 0 < where Re h(z) > 0. 2r l-r 2 Aso 2 1 2 IH(z)h(z) + zh’ (z)Id8 < < 2 2 ]H(z)h(z)[dO + Izh’ (z)Id6) (l+(k2-1) r2)1/2 (i+3r2) 1/2 l-r 2 + 2r l-r (2.8) 2 by using Schwarz’s inequality, lemma 2 and (2.7). Hence from (2.6) and (2.8), we have (n+l) 2an+I 1 21/2k nl<_ rnfl 2 -n a [(l+(k2 -i) r2 1/2+i] 1 (l-r) 1/2k+l K.I. NOOR 332 2 and so choosing II fan+ll-lanl n we obtain for r l+(k2-1)r < + n>_l 1 n \3! Thus 1/2k-I <_ c(k)n lan+ll-lanll (}- I) The function F defined by (2.1) shows that the index k We now evaluate the radius of convexity for the class T THEOREM 5: Let feT k. is best possible. k. Then the radius R of the circle which f maps onto a convex domain is given by 1/2k+2>-X/k)], R The function F k defined by (2.1) shows that this result is best possible. 2, R ticular, when k 2aV--,which In par= This result also follows from is well known. the remark in [3,p.23]. PROOF: By definition ag’ (z)h(z) zf’(z) geVk; Re h(z)>O. Thus (zf’ (z))’ f’(z) (z) + zh’ h(z) (zg’ (z))’ g’(z) and so (zf’ (z))’ f’(z) For geVk, it is well known Re (z)) Re (zg > izh’(z) hz) g’ (z) [9] that, for z (zg’ (z)) g’ (z) > re i8 0<r<l, r2-kr+l l_r 2 Hence Re (zf’ (z)) f’(z) > r2-kr+l l-r 2 2r l-r 2 2 r -(k+2)r+l 2 l-r This gives the required result. (i). We also note that the extremal function function as FB(z) Fk(Z) defined by equation (2.6) in [3]. defined by (2.1) is the same As A. W. Goodman has pointed out that this function is sometime referred to as the generalized Koebe function. GENERALIZATION OF CLOSE-TO-CONVEXITY (li). 333 We conjecture that the class T is a proper subclass of the class K(B) as k defined in [3], since in the definition of Tk, gV and we know that k gVk, 2<_k<__4, is convex in one direction and all the functions in one direction form a proper subclass of the class of close-to-convex functions. (ii). It remains open whether T ACKNOWDGEMENT. k is a linear in variant family. The author wishes to thank Prof. A. W. Goodman and the referee for their interesting and helpful comments, which influenced the final version of this paper. REFERENCES i. HAYMAN, W. K. Multlvalent Functions, Cambridge University press, 1967. 2. KAPLAN, W. Close-to-convex Schilcht Functions, Mich. Math. J. 1 (1952), 169-185. 3. GOODMAN, Ao W. On Close-to-convex Functions of Higher Order, E6tvos. Sect. Mathematica, 15 (1972), 17-30. 4. BRANNAN, D. A. On Functions of Bounded Boundary Rotation, Proc. Edln. Math. Soc. 2 (1968/69), 330-347. 5. PAATERO, V. Uber Geblete von beschrankter Randdrehung Annal/Acad. Scl. Fenn. Set A 37 (1933), 20pp. 6. GOLUSIN, G. M. 7. POMMERENKE, Ch. On Starlike and Close-to-convex Functions, Proc. Lond. Math. Soc. 3 (1963), 290-304. Univ. Sci. Budapest. Geometrische Functionstheorie, Berlin, 1957.