Computers and Mathematics with Applications

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1 Compuers and Mahemaics wih Applicaions 6 () 6 63 Conens liss available a ScienceDirec Compuers and Mahemaics wih Applicaions journal homepage: Cerain resuls for a class of convex funcions relaed o a shell-like curve conneced wih Fibonacci numbers Jacek Dziok a, Ravinder Krishna Raina b,, Janusz Sokół c, a Insiue of Mahemaics, Universiy of Rzeszów, ul. Rejana 6A, 3-3 Rzeszów, Poland b M.P. Universiy of Agriculure and Technology, Udaipur, India c Deparmen of Mahemaics, Rzeszów Universiy of Technology, ul. W. Pola, 3-99 Rzeszów, Poland a r i c l e i n f o a b s r a c Aricle hisory: Received 3 June Received in revised form March Acceped March Keywords: Univalen funcions Convex funcions Sarlike funcions Subordinaion Fibonacci numbers Trisecrix of Maclaurin Conchoid of de Sluze This paper invesigaes some basic geomeric properies for he class KSL of funcions f analyic in he open uni disc = {z : z < } (which is relaed o a shell-like curve and associaed wih Fibonacci numbers) saisfying he condiion ha f () =, f () = and zf f τ z + τ z τ z τ z where, he number τ = ( )/ is such ha τ fulfils he golden secion of he segmen [, ]. Some relevan remarks and useful connecions of he main resuls are also poined ou. Elsevier Ld. All righs reserved.. Inroducion Assume ha A is he class of all holomorphic funcions f in he open uni disc wih normalizaion f () =, f () =, and le S (as is cusomary) be he subclass of A which consiss of univalen funcions. We say ha f is subordinae o F in, wrien as f F, if and only if f = F(ω) for some holomorphic funcion ω such ha ω z, z. The class SL of shell-like funcions is he se of funcions f A saisfying he condiion ha where zf f p (.) p = + τ z τ z τ z, τ =.68, (.) I should be observed ha SL is a subclass of he class of he sarlike funcions S. The class of shell-like funcions SL was defined in [] and furher examined in []. The funcion (.) has some nice properies. The name aribued o he class SL is moivaed by he shape of he curve C = p(e i ), [, π) \ {π}, (.3) Corresponding auhor. addresses: jdziok@univ.rzeszow.pl (J. Dziok), rkraina_7@homail.com (R.K. Raina), jsokol@prz.edu.pl, jsokol@prz.rzeszow.pl (J. Sokół). Presen address: / Ganpai Vihar, Opposie Secor, Udaipur 33, Rajashan, India. 898-/$ see fron maer Elsevier Ld. All righs reserved. doi:.6/j.camwa..3.6

2 66 J. Dziok e al. / Compuers and Mahemaics wih Applicaions 6 () 6 63 which is a shell-like curve and a simple ransformaion convers i ino a curve called he conchoid of de Sluze (René François Baron de Sluze 6 68). For deails, see Secion. Moreover, he coefficiens of (.) are conneced wih he Fibonacci numbers as explained in Lemma. and in he nex secion. A geomeric descripion of he conchoid of de Sluze is given here as follows: A ray OB is drawn from he poin O(, ) and i cus he direcrix x = a, where a > a he poin B(a, b). From he poin B(a, b), segmens BM and BN are laid off in eiher direcion along he ray such ha OB BM = k and OB BN = k, where k > is given. As b changes, he ray revolves, and he poin M describes a curve (called he conchoid of de Sluze) given by a(x a)(x + y ) + k x =, while he poin N describes a curve (called he conjugae of he conchoid of de Sluze) given by a(x a)(x + y ) k x =. Moivaed by hese ideas, we consider for he purpose of his paper a cerain class KSL of funcions analyic in he open uni disc which connecs (by means of he subordinaion) he class K of convex funcions wih he class SL of shelllike funcions. The various resuls sudied depic some of he basic geomeric properies for his funcion class KSL. Some relevan cases and useful remarks are also menioned. Definiion. The funcion f A belongs o he class KSL of convex shell-like funcions if i saisfies he condiion ha + zf f p where he funcionp is defined in (.). I may be poined ou here ha Ma and Minda in [3] defined C(ϕ), (or S (ϕ)) o be he class of all normalized funcions f = z + a z + such ha + zf /f ϕ, (or + zf /f ϕ), where ϕ is fixed funcion analyic and univalen in he uni disc wih Reϕ >, ϕ( ) is symmeric wih a real axis and ϕ is sarlike wih respec o ϕ() = and ϕ () >. Several subclasses of convex and sarlike funcions were unified in his way. Because he funcionp is no univalen in, we canno use he resuls from [3] o obain some heorems on he class KSL. The radius of univalence and several oher properies of he funcionp were found in [4]. Le us recall some of hem. Lemma. ([4]). Le he funcionp be given by (.), hen i saisfies he following: () p is univalen in he disc z < (3 )/.38, any increase in he greaer side makes he asserion false, () p = + n= (u n + u n+ )τ n z n = + τ z + 3τ z + 4τ 3 z 3 + 7τ 4 z 4 + τ z +, where {u n } is he sequence of Fibonacci numbers u =, u =, u n+ = u n + u n+ (n =,,, 3,...), (3) lim ϕ π Im[p(e iϕ )] =, and lim ϕ π + Im[p(e iϕ )] =, (4) Rep(e iϕ ) = (3 cos ϕ) = γ for all ϕ [, π). In [4], he auhors presened a class SLM α, α [, ], of funcions ha are analyic in he open uni disc such ha [ f () =, f () = and α + zf ] + f ( α) zf p( ) for all z. f This class SLM α is relaed o he presenly invesigaed class KSL only hrough he funcionp and SLM α KSL for all α. I easy o see ha KSL SLM bu KSL SLM becausep is no univalen funcion. The presen paper deals wih ideas and echniques used in geomeric funcion heory. The cenral problem considered here is he coefficien esimaes for his class depiced by he Fibonacci numbers. Besides he coefficien problems, we also provide some ineresing corollaries concerning he connecions of our defined class wih oher well known classes.. Preliminary lemmas For some in-deph undersanding of he class KSL i would be worhwhile here o find he shape of he curve C = {p(e i ), [, π) \ {π}}. We begin our sudy by noing ha p() = p = and p() = p(τ 4 ) = τ. Moreover, because p e ±i arccos(/4) = = γ, (.4) (.) (.6)

3 J. Dziok e al. / Compuers and Mahemaics wih Applicaions 6 () Fig.. Curve C : (x )y = ( x)( x ). so he curve C inersecs iself on he real axis a he poin γ. If we denoe Rep(e iϕ ) = x and Imp(e iϕ ) = y, ϕ [, π) \ {π}, hen afer simple calculaions, we ge x = (3 cos ϕ), y = sin ϕ(4 cos ϕ ), ϕ [, π) \ {π}. (.) (3 cos ϕ)( + cos ϕ) I is useful here o use (.) o find he corresponding Caresian equaion of he curve C. This curve is described by he equaion (x )y = ( x)( x ). (.) I is worhy o poin ou ha for k = a, he conchoid of de Sluze (.4) becomes he risecrix of Maclaurin (Colin Maclaurin ): x 3 + 3ax + (x a)y =, (.3) while he conjugae of he conchoid (.) becomes he conjugae of he risecrix of Maclaurin given by x 3 ax + (x a)y =. (.4) If we rewrie (.) in he following form 3 x + 3 x + x y =, hen he image of he uni circle under he funcionp is ranslaed ino a risecrix of Maclaurin (.3) (wih a = ( τ)/ = /). Therefore he curve C has a shell-like shape, see Fig.. Le us refer he class K(α) S, ( α < ), of convex funcions of order α, inroduced in [] and defined by [ K(α) = f A : Re + zf f ] > α for all z. If f KSL, hen by (.6) we have [ Re + zf ] > f min {Rep, z } > by condiion (4) of Lemma.. Therefore, we obain KSL K(γ ), (.) where γ = /.36, which means ha if f KSL, hen i is convex of order γ and hence univalen in he uni disc. Corollary.. If f is in KSL, hen f is univalen in.

4 68 J. Dziok e al. / Compuers and Mahemaics wih Applicaions 6 () 6 63 Le us recall he relevan connecion of he funcion defined by (.) wih he Fibonacci numbers u n = ( τ)n τ n, τ =, (n =,,, 3,...) (.6) conained in () of Lemma.. Moreover if p = + p n z n, n= hen he coefficiens p n saisfy τ for n =, p n = 3τ for n =, τ p n + τ p n, for n = 3, 4,,..., where τ =. I is also worh noing here ha since τ = τ τ, so he number τ divides [, ] such ha i fulfils he golden secion of his segmen. Lemma. ([]). If a funcion (.7) f = z + a n z n (z ) is in he class SL, hen a n τ n u n n =, 3, 4,..., (.8) (.9) where u n are given in (.6). This resul is sharp and he equaliy in (.9) is aained by he funcion z g = τ z τ z = [ ] τ + z = z + u τ n τ n z n (.) z 3. Main resuls Corollary 3.. If a funcion f of he form (.8) belongs o he class KSL, hen a n τ n u n n n =, 3, 4,..., where u n are given in (.6). This resul is sharp and he equaliy in (3.) is aained by he funcion (3.) f = + τ log + z τ z (3.) Proof. A funcion f is in he class KSL if and only if here exiss a funcion g SL such ha g = zf for all z, (3.3) or equivalenly f = g() d for all z. The relaions (3.3) and (3.4) follow direcly from (.) and (.6). Therefore, if zf = z + na n z n (z ) (3.4) (3.)

5 J. Dziok e al. / Compuers and Mahemaics wih Applicaions 6 () belongs o he class SL, hen by Lemma., we conclude ha na n τ n u n, which esablishes (3.). The funcion (3.) is such ha z f = g, where he funcion g is given in (.), and hence by (3.3), i follows ha f KSL. Moreover, by (.) we ge u n τ n f = z + Thus he resul (3.) is sharp. n z n Theorem 3.. A funcion f belongs o he class KSL if and only if here exiss an analyic funcion q, q p, such ha [ w ] q() f = exp d dw. (3.6) (3.7) Proof. I is known ha a funcion g SL can be expressed by g = z exp q() d, where q p. In view of (3.3), we infer ha f KSL if and only if zf = g (wih g SL), and inegraing (3.8), we obain (3.7). Theorem 3. provides us a mehod of finding he members of he class KSL. Le us refer he class S (α)( α < ) of sarlike funcions of order α, inroduced in [] and defined by S (α) = f A : zf + ( α)z f z and he class (see [6,7]) S (A, B) = where < B < A. f A : zf f + Az + Bz, Theorem 3.3. If a funcion f belongs o he class KSL, hen here exiss a funcion g S (, τ ) and a funcion h S (/) such ha (3.8) z f = gh (3.9) Proof. Le f KSL, hen by Theorem 3., here exiss a holomorphic funcion ω wih ω() = and ω < for z such ha z f = z exp Observe ha p (ω()) = p (ω()) d. (3.) τ ω() +, (3.) + ω() so we can rewrie (3.) in he form z f = z exp = z exp = : gh. τ ω() τ ω() d + d z exp +ω() d +ω() Using he srucural formulas for he class S (A, B) (see [7, p. 3]) and for he class S (α) (see [8, p. 7]), we can find ha he funcions g and h defined by (3.) saisfy g S (, τ ) and h S (/), respecively. d (3.)

6 6 J. Dziok e al. / Compuers and Mahemaics wih Applicaions 6 () 6 63 Fig.. The curve D = { f (e iϕ ), ϕ [, π) \ {π}}. If he funcions g S (, τ ) and h S (/) are generaed by heir srucural formulas wih he same funcion ω, hen we can reverse he above seps of he proof, and by (3.) and (3.), we can ge he funcion f such ha z f = gh (z ) is in he class KSL. For example, if ω = xz, x, hen (3.) gives z z f = ( + xz)( τ xz) Upon inegraing we obain ha he funcions f = + xz log + τ τ xz belong o he class KSL for all x. For x = i becomes he funcion f given by (3.), which shows he sharpness of he esimaion (3.) of coefficiens in he class KSL, for is Taylor expansion see (3.). This funcion is also exremal for he oher problems in his class. Le us see wha i looks like f ( ). To find he image of under he funcion f = + τ log + z τ z we observe ha he funcion w = + z τ z (z ) maps he circle z = ono he circle w R = R, where R =.36. If we denoe Argw(e i ) = ψ(), [, π), hen i is easy o observe ha ψ() ( π/, π/) and w(e i ) = R cos ψ(). By seing Re f (e i ) = x and Im f (e i ) = y, [, π) \ {π}, hen afer some calculaions, we ge from (3.) ha x = + log[r cos ψ], y = + ψ, ψ π, π. (3.3) Consequenly, he funcion (3.) maps he uni circle ono a curve D described by (3.3), see Fig.. The funcionsp and f are conneced by he relaion: + z f f = p Jus as he Koebe funcion plays a cenral role in he class S he funcion f plays a cenral role in he class KSL. Theorem 3. provides us a mehod of finding he members of he class KSL for given funcion q, q p. Much more hard o verify is he quesion if given f belongs o he class KSL. Even when we consider a simple polynomial, e.g. g = z +cz n, hen (.6) becomes + n cz n p (z ) + nczn and i is difficul o find all c saisfying his subordinaion because p is no univalen. The nex heorem excludes some polynomials of KSL and somewha solves his problem.

7 J. Dziok e al. / Compuers and Mahemaics wih Applicaions 6 () Theorem 3.4. If n {, 3, 4,...} and c > n( n ), (3.4) hen he funcion g = z + cz n does no belong o he class KSL. Proof. Le us denoe G := + zg g = + n cz n + ncz n We prove ha if (3.4) is saisfied, hen G p. I suffices o show ha G( ) p( ). The sep( ) is on he righ of he curve in Fig.. The se G( ) is a disc wih he diameer from x = n c o x n c = +n c. If (3.4) is saisfied, hen he +n c one of x i, where i =,, saisfies x i < γ = /, and hen G( ) p( ). This proves Theorem 3.4. Thus for n = : if g = z + cz KSL, hen c ( )/(4 ).8. Theorem 3.. If f KSL( z = r, r < ), hen and ( + r) γ γ f f ( r) = + τ log + τ r r (3.) + ( + τ )r + τ r f f ( r) = where γ = / and τ = ( )/. The upper bounds are sharp. + τ r τ r, (3.6) Proof. Le f KSL be of he form (.8), and le us denoe by b n he coefficiens of he funcion f given by (3.6). Then by (3.), we have a n b n for each ineger n. Noice ha he even coefficiens b n = u n τ n /n are negaive, while he odd coefficiens are posiive. If z = re iθ, hen from he coefficien inequaliy a n b n, we obain f r + a n r n r + b n r n = r + b n ( ) n r n = f ( r) which yields he upper bound of (3.). Analogously, we can obain f = + na n z n + n a n z n + n b n z n = + n b n r n = b r + 3b 3 r 4b 4 r 3 + = f ( r), and we ge he upper bound of he inequaliy (3.6). I is easy o see ha he upper bounds are sharp, being aained by he funcion f KSL a he poin z = r. To find he lef-hand side of he inequaliy (3.), le us recall (see [7, pp. 3 37]), ha if g S (, τ ), hen for z = r ( r < ), we have r + τ r g. Moreover, if h S (/), hen for z = r ( r < ), we ge r + r h. (3.7) (3.8) By Theorem 3.3, we have z f = g h wih g S (, τ ) and h S (/), and muliplying he respecive sides of (3.7) wih hose of (3.8), we hus obain he lef-hand side of (3.6). To prove he lef-hand side of he inequaliy (3.), we noe ha by (.) if f KSL, hen f K(γ ) is convex of order γ. The desired inequaliy will now follow from he well-known inequaliy for f wih f K(γ ) (see, for example [8, p. 39]).

8 6 J. Dziok e al. / Compuers and Mahemaics wih Applicaions 6 () 6 63 Theorem 3.6. If a funcion f of he form (.8) belongs o KSL, hen n a n 3b 3 = 3.764, n =, 3, 4,..., (3.9) where b n = u n τ n /n are he coefficiens of f KSL given by (3.6). Moreover, if he coefficien sequence {an } of he funcion f converges, hen lim n a n +.7, n wih equaliy holding for he coefficiens b n of he funcion f. Proof. If f = z + b z + b 3 z 3 +, hen by (3.6) we have nb n = τ n n + n n u n =, (3.) which implies ha and where nb n = (n )b n τ + (n )b n τ b n b n 3b 3 for n, b = (( )/4).3, b 3 = (3 )/3.6. Using (3.), we have a n b n, which readily yields he inequaliy (3.9). The coefficien inequaliy a n b n and he limiing case ha + lim n b n = n leads a once o he oher inequaliy (3.). By puing z = re iϕ, ϕ [, π) \ {π}, and performing simple calculaions, we ge p re iϕ = + τ r e iϕ τ re iϕ τ r e iϕ = ( + τ r )( τ r τ r cos ϕ) + iτ r( τ r + 4τ r cos ϕ) sin ϕ. τ re iϕ τ r e iϕ τ re iϕ τ r e iϕ Hence Im[p re iϕ ] Re[p re iϕ ] = τ r( τ r + 4τ r cos ϕ) sin ϕ ( + τ r )( τ r τ r cos ϕ) τ r( τ r 4τ r) := φ(r), (3.) ( τ r + τ r)( + τ r ) whenever r < r = 3. By Lemma. for such r he curvep(re i ), [, π) \ {π}, has no loops andp is univalen in r = {z : z < r }. Therefore [ zf f p, ] [ zf z r f ] p r for all z r, where he subordinaion F G in a disc z < r denoe ha F = G(ω) for some holomorphic funcion ω, ω() = and ω < for all z < r. A simple geomeric observaion of (3.) gives he following heorem. Theorem 7. If f KSL, hen [ Arg + zf ] < f arcan φ(r) z < r < r = 3, (3.) where φ(r) is given by (3.). This heorem says ha if f KSL, hen f is srongly convex of order arcan φ(r) (see [9]) in he disc z < r, whenever π r < r.

9 J. Dziok e al. / Compuers and Mahemaics wih Applicaions 6 () Acknowledgemens The auhors would like o express heir sinceres hanks o he referees for a careful reading and various suggesions made for he improvemen of he paper. References [] J. Sokół, On sarlike funcions conneced wih Fibonacci numbers, Folia Sci. Univ. Tech. Resoviensis 7 (999) 6. [] J. Dziok, R.K. Raina, J. Sokół, On a class of sarlike funcions relaed o shell-like curve conneced wih Fibonacci numbers (submied for publicaion). [3] W. Ma, D. Minda, A unified reamen of some special classes of univalen funcions, in: Proceedings of he Conference on Complex Analysis, Tianjin, 99, in: Conf. Proc. Lecure Noes Anal., vol. I, Inernaional Press, Cambridge, MA, 994, pp [4] J. Dziok, R.K. Raina, J. Sokół, On α-convex funcions relaed o shell-like funcions conneced wih Fibonacci numbers, Appl. Mah. Compu.,, in press (doi:.6/j.amc...9). [] M.S. Roberson, Cerain classes of sarlike funcions, Michigan Mah. J. 76 () (94) [6] W. Janowski, Exremal problems for a family of funcions wih posiive real par and some relaed families, Ann. Polon. Mah. 3 (97) [7] W. Janowski, Some exremal problems for cerain families of analyic funcions, Ann. Polon. Mah. 8 (973) [8] A.W. Goodman, Univalen Funcions, vol. I, Mariner Publishing Co., Tampa, Florida, 983. [9] J. Sankiewicz, Quelques problèmes exrémaux dans les classes des foncions α-angulairemen éoilées, Ann. Univ. Mariae Curie-Skłodowska Sec. A (966)