SUBCLASSES OF CLOSE.TO-CONVEX FUNCTIONS

Transcription

1 It. J. ath. & Math. Sci. Vol. 6 No. 3 (1983) SUBCLASSES OF CLOSE.TO-CONVEX FUNCTIONS E.M. $1LVIA Departmet of Mathematics Uiversity of Califoria, Davis Davis, Califoria (Received lauary 4, 1983) ABSTRACT. Let [C,D], -1 (_ D ( C i, deote the class of fuctios $(z), S(0) S (0) I 0, aalytic i the uit disk U (z: Jz < 1] such that I + (zg (z)/g (z)) is subordiate to (l+cz)/(1+dz), z s U. We ivestigate the subclasses of close-to-covex fuctios f(z), f(0) f (0) I 0, for which there exists S s }([C,D] such that f /g is subordiate to (l+az)l(l+bz), -I (_ B ( A I. Distortio ad rotatio theorems ad coefficiet bouds are obtaied. It is also show that these classes are preserved uder certai itesral operators. WORDS AND PHRASES, Uivalet, covex, starlike, subordiatio, covolutio. AMS (MOS) SUBJECT CLASSIFICATION (1980) CODES: 30C45, 30C55. 1 INTRODUCTION, Let S deote the class of fuctios f(z) z + a z aaly --. tic ad uivalet i the uit disk U (z: ]z I). For fuctios ad G aalytic i U we say that S is subordiate to G, deoted < G, if there exists a Schwarz fuctio w(z), w(z) aalytic i U with w(0) 0 ad ]w(z) ( I i U, such that g(z) G(w(z)). If G is uivalet i U the S G if ad oly if (0) G(0) ad s(u) c

2 450 E. M. SILVIA G(U). For A ad B, -1 <_ B A _< 1, fuctio p aalytic i with p(o) I is i the class [A,B] if p(z) (l+az)/(l+bz). This class was itroduced by Jaowski [4]. Give C ad D, -I i D < C I, ][C,D] ad J [C,D] deote the classes of fuctios f alyric i U with f(0) f (0) -I 0 such that I + zf (z)/f (z) s [C,D] ad zf (z)/f(z) s [C,D], respectively. The classes JS[C,D] were itroduced by Jaowsci [4] ad studied further by Goel ad Mehrok ([1] ad [3]). For C i ad D -I, }[i,-i] ( (,/ [I,-I],/ ), the well-kow subclass of covex (starlike) fuctios. A fuctio f(z) z + a z aalytic i --. U is said to be i the class C[A,B;C,D], -I <_ B < A <_ I, -I <_ D C! I, if there exists g s ){[C,D] such that f /g e.[a,b]. The well-kow (Kapla [5]) class of close-to-covex fuctios is C[I,-I;I,-I] C while ad [A,B] c [i,-i] shows C[A,B;C,D] C C c S. Sice, z g s,/ [C,D] if ad oly if g() 0-1 d e [C,D], we also ote that C[I,-I;C,D] was studied by Goel ad Mehrok ([] ad [3]). I Sectio Of this paper we obtai distortio ad rotatio theorems for f (z) wheever f s C[A,B;C,D] ad a subordiatio result relatig C[A,B;C,D] ad [A,B]. I Sectio 3, it is show that the class C[A,B;C,D] _ is preserved uder certai itegral operators. We coclude with coefficiet iequalities.. DISTORTION AND ROTATION THEOREMS. Uless otherwise metioed i the sequel, the oly restrictios o the real costata A, B, C ad D are that -i D < C I ad -1.i B < A.i 1. THEOREM 1. f s C[A,B;C,D], Izl i r < 1, (1-At) (1-Dr) (C-D)/D < if (z)l < 1-Br (l+ar)(l+dr) (C-D)/D D 0 I+Br

3 SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS 451 ad (-A) x,-cr) l-br < If (z)l < 1+Ar)ex{Cr) D O. Th e bouds are sharp. PROOF. For f 8 C[A,B;C,D], there exists a g 8 [C,D] ad p e [A,B] such that Sice g e [C,D] if ad oly if zg e " [C,D], for z! r i [4] (1-Dr) (C-D)/D! Ig (z)l -- exp(-cr) <_ Ig (z)l _< exp(cr) For p z [A,B], Iz] _ r, f (z) g (z)p(z). (.1) (C-D)/D (l+dr) D 0, ad (.) the uivalece of (I+Az)/(I+B) gives l-at < Ip( -)l +A (.) I-Br I+Br The result follows immediately upo applyig (.3) ad (.) to (.1). Equality is obtaied for f e C[A,B;C,D] satisfyig _l+az) (1+Dz) (C-D)/D I + Bz D 0 f (z) I+Az) exp(cz) I + Bz D 0 (.4) ad z + r. REMARK. For A 1 ad B -1, Theorem 1 agrees with Theorem 3 of Gee1 ad Mehrok [].

4 45 E. M. SILVIA THEOREM. For f, CtA,B;C,D], Izl <-- r < I, Jars f (z) arcsi(dr) + arcsi (A-B); D 0 I_ABr (A-B)r D 0 arcsi(cr) + arcsi I-ABr PROOF. From (.1) we have Sice zg e [C,D], we kow [] that for ]zl _ r < 1 (.5) arcsi(dr) D 4 0 (.6) For p s [A,B], P(IZl < r) is cotaied i the disk I" --I-AB;1 < (A-B)r frm which it f11 that 1-Br 1-B r larg P(z)! arcsi (A-B)r " I-ABr (.7) Substitutig (.6) ad (.7) ito (.5) gives the result. REMARKS 1. For A 1, B -1, Theorem agrees with Theorem 4 of Goel ad Mehrok [].. For A C 1, B D -1, Theorem reduces to the result of Krzyz [7] that ]arg f (z)]! (arcsi r + arcta r),

5 SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS 453 The covolutio or Hadamard product of two power series f(z) a z ad g(z) b z is defied as the power series =O =O (f g) (z) a b z =O I order to obtai a subordiatio result likig C[A,B;C,D] ad [A,B] we eed the followig ad b_e covex i U ad suppose f <. The LEMMA A (Ruscheweyh ad Shell-Small, [11]). THEOREM 3. I f f C[A,B;C,D] the there exists p e [A,B] such that for all s ad t with I+Dsz!C-D)/D () D 0 f (sz)p(tz)_ f (tz)p(sz) (.8) exp[c(s-t)z] D 0. PROOF. We will use a approach due to Ruscheweyh [10]. From (.1) we have zf (z zg (z) + zv (z) f (z) g (z) for g e p(z) [C,D] ad p s [A,B]. Therefore, zf z.l_ z (z) f (z) (1 + zx"(z) p(z) 1 < g (z) ( 1 + Dz 9) For s ad t such that Isl I, tl! I, the fuctio Z h(z) (l-su 1-tu )du is covex i U. Applyig Lemma A to (.9) with this h, we have (zf (z) zp (z)) * h(z) < (C-D)z. h(z) f (z) p(z) 1 + Dz (.10) Give ay fuctio (z) aalytic i U with (0) O, we have

6 454 E. M. SILVIA ( h)(z) SZ tz du (u) z 8 U, so that (.10) reduces to lg(p(sz)f (sz)v(tz))-) (tz < (C-D) SZ du 1 tz + Du (.11) Itegratig the righthad side of (.11) ad expoetiatig both sides leads to (.8). COROLLARY 1. If f 8 C[A,B;C,D] the there exists a_ p e f[a,b] ad a Schwarz uctio w(z) such that (C-D)/v p(z) (1 + Dw(z)) D 0 f (z) p(z)exp(cw(z) PROOF. The result follows directly upo substitutig s 1 ad t 0 ito Theorem 3. COROLLARY. If f(z) z + a z e C[A,B;C,D] the = (C-D) + (A-B) 1"1! PROOF. If g < F the Ig (O) [8] From Corollary I, we take g(z) f (z)/p(z) ad F(z) l+dz) (c-d)/d D 0 xp(cz) D 0 The g (0) a c I for p(z) 1 + c z ad F (0) C D. =l (C-D) +I Cll (C-D) + (A-B) Therefore 1a1- l1{ <- {C-D{ ad lal <_ <- as claimed.

7 SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS INVARIANCE PROPERTIES. We will eed the followis lemmas. LEMMA B (Ruscheweyh ad Shell-Small, [ii]). ad g starlike i U. The for F aalytic i *Fg(u) is cotaied i the covex hull of,g Le_._t q b_e covex U with F(0) 1. F(U). LEMMA C (Silverma ad Silvia, [1]). e /[C,D] the for s, *g./ [C,D]. THEOREM 4. l f cp K, ad f e C[A,B;C,D] the 0*f s C[A,B;C,D]. PROOF. For f s C[A,B;C,D] there exists g / [C,D] ad F s [A,B] such that zf (z) g(z)f(z). Sice (l+az)/(l+bz) is covex i U, by Lemma B, for 0 v. From 0*f e C[A,B;C,D]. z(0sf) 0*Fg.< I+Az o,g o,g l+bz (z e U) (3.1) Lemma C, 0,g e *[C,D] so that (3.1) is equivalet to REMARK. For A C i, B D -1, Theorem 4 was proved by Ruscheweyh ad Sheil-Small [11]. COROLLARY. I.f f e C[A,B;C,D] the so are ad (i) V l(z) rjz tt-if(t)dt Re 7 > 0 z 7 0 (ii) F (z) z f(_)-f(x.) d, x 1 1. o -x

8 456 E. M. SILVIA 1 l-x PROOF. Observe that F. (z) (h.*f) (z), j 1,, where J i + z Re > 0, ad (z) --1 (l-x) log[ l-xz], Ix i, x I. Sice h I was show to be covex, l-z by Ruscheweyh [9] ad h is clearly covex, the result follows immediately from Theorem 4. REMARK. Goel ad ]/ehrok [3] showed that C[1,-1;C,D] was preserved uder Fl(Z) whe 7 1,,3 ad uder F(z) by a differet method. whe x =-1 b1 _ --- C-D ad 4f,. COEFFICIENT INE(UALITIES We begi with coefficiet iequalities for ][C,D]. LEblt/A. For g(z) z + b z s E[C,D] ad Ix complex ; PROOF. For g(z) z + bz -- e- - - tu,vj, there exists a Schwarz fuctio w(z) such that 1 + (zg (z)/g (z)) (l+cw(z))/(l+dw(z)) or zg (z)/g (z) (C-D)w(z)/(l+Dw(z)). C-D C-D b- 71 ad b3-6 (7 + (C-D)71 }" -- Therefore, Ib C-D ]! ad C-D b3 Ixb 6 { + (C-D) -", Ix(C-D) ]71 ). (4.1) Substitutio of the series expasios ad compariso of the coefficiets leads to

9 SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS 457 We kow [6] that for s complex (4.) Combiig (4.) ad (4.1) yields the result. REMARK. If we apply the iequality [8] to (4.1), the same proof shows that [ + Ib3-11b I(C-D)-,(C-D) i} Ibl THEOREM 5. For f(z) z + a z s C[A,B; C,D] =,la1 (C-D)+(A-B) ad + (A-B)(C-D+1) C-D[ I (C-D) (C-D) + (A-B) (C-D+1) C-D 1 PROOF. There exists a g(z) z + b z R[C,D] ad a Schwarz = fuctio w(z) z such that f (z)ig (z) (l+aw(z))l(l+bw(z)) =l A-B z e U. Comparig series expasios, we see a^ b +--:-- i ad a3 b3 + (A_B)b71 + (A-B) ( -By 3 1 )" (4.3) The boud for a follows from the Lemma. Applyig (4.) ad the Lemma (p 0) to (4.3), we have

10 . 458 E. SILVIA C-D (A-B) 3 max(l, IB[ C-D REFERENCES Goel, R. M. ad Mehrok, B. S., O the coefficiets of a subclass of starlike fuctios. Idia 7. Pure Awl. Math.m 1(1981), Goel, R. M. ad Mehrok, B. S., O a class of close-tocovex fuctios. Idia J. Pure Appl. Math., 1(1981), Goel, R. M. ad Mehrok, B. S., Some ivariace properties of a subclass of close-to-covex fuctios. Idia J. Pure Awl. Math. 1(1981), Jaowski, W., Some extremal problems for certai families of aalytic fuctios. A. Polo. Math.m 8(1973), Kapla, W., Close-to-covex schlicht fuctios. Math. J., 1(195), Mich Keogh, F. R. ad Merkes, E. P., A coefficiet iequality for certai classes of aalytic fuctios. Proc. Amer, Math, Soc.m 0(1969), 8-1. Krzyz, J., O the derivative of close-to-covex fuctios. Colloq. Math.m 10(1963), Nehari, Z., Coforms1Mavvi. McGraw-Hill, New York, 195. Ruscheweyh, St., New criteria for uivalet fuctios. Proc. Amer. Math. Soc.m 49(1975), Ruscheweyh, St., A subordiatio theorem for -like fuctios. J. Lodo Math. Soc., 13(1976), Ruscheweyh, St. ad Sheil-Small, T., Hadamard products of schlicht fuctios ad the Poly Schoeber8 cojecture. Comm. Math. Helv. 48(1973), Silverma, H. ad Silvia, E. M., Subclasses of starlike fuctios subordiate to covex fuctios (submitted).

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