LINEAR INVARIANCE AND INTEGRAL OPERATORS OF UNIVALENT FUNCTIONS

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1 LINEAR INVARIANCE AND INTEGRAL OPERATORS OF UNIVALENT FUNCTIONS MICHAEL DORFF AND J. SZYNAL Absrac. Differen mehods have been used in sudying he univalence of he inegral ) α ) f) ) J α, f)z) = f ) d, α, R, where f belongs o one of he known families of holomorphic and univalen funcions fz) = z + a 2 z 2 + in he uni disk D = {z : z < } see [5]). In his paper, we sudy a larger se han ), namely he se of he minimal invarian family which conains ), where f belongs o he linear invarian family, and hereby we obain informaion abou he univalence of ). In paricular, we deermine he order of his minimal invarian family in he cases of univalen and convex univalen funcions in D. As a resul, we find he radius of close-o-convexiy and he lower bound for he radius of univalence for he minimal invarian family in he case of convex univalen funcions. This allows us o deermine he exac region for α, ) where he corresponding minimal invarian family is univalen and close-o-convex. These resuls are sharp and generalize hose which were obained in [].. Inroducion Le S denoe he class of holomorphic and univalen funcions f in he uni disk D = {z : z < } which have he form 2) fz) = z + a 2 z 2 +, z D, and le S c S be he subclass consising of convex funcions. These wo classes are examples of linear invarian family as described by Pommerenke [2]. If M is a linear invarian family of locally univalen funcions of he form 2), hen he order of such family is defined by 3) ord M = sup a 2. f M We have ord S = 2 and ord S c =. For α, R and f S ors c), consider he inegral operaor ) α ) f) 4) F α, z) = J α, f)z) = f ) d, z D. Properies such as univalence, convexiy, and close-o-convexiy, of he funcion F α, have been inensively sudied see [5] for references). The following hree operaors are of paricular ineres and heir univalence in he case of f S) have no ye been seled:

2 2 MICHAEL DORFF AND J. SZYNAL a) he Biernacki inegral, F,, was claimed by Bernacki o be univalen, bu his claim was disproved by a counerexample by Krzyz and Lewandowski [7]. The exac radius of univalence for F, is no known. Lewandowski [8] has given he bes known esimae for r u F, ) >.9. b) he Royser-Pfalzgraff inegral, F α,, is known o be univalen for α /4 α C) [] and o be nonunivalen if α > /3, α [4]. The lower bound for he radius of univalence, r u F α, ), is given in [3]. c) he Danikas-Ruscheweyh inegral, F,, has been conjecured recenly o be univalen [3]. Also, he operaor, J, 2 f), f S c, is imporan and appears in he paper of Hall [6]. In his paper we firs deermine he order of he minimal invarian families M α, S) and M α, S c ) conaining he ses 5) J α, S) = {F α, : f S}, J α, S c ) = {F α, : f S c }, respecively, where F α, is given in 4). From his, we obain he sharp value of he radius of close-o-convexiy and he bound for he radius of univalence for he family M α, S c ), which allows us o deermine he exac region for α, ) in which J α, f), f S c is univalen and even close-o-convex) in D. The las resul generalizes he earlier resuls abou univalence of J α, and J,, f S c []. We will use following lemmas. 2. Background Resuls Lemma A. Prokhorov, Szynal [3]) Le M be a linear invarian family of funcions f of he form 2) such ha fz)/z, z D and le J α, M) denoe he se of funcions F α, given by ) ξ z 6) G α, z) = Ĵα,f)z) f )) α ) f) = d. ξ) 2 2α ξ Then he family M α, of funcions G α, is he minimal invarian family conaining he class J α, M), where ξ is an arbirary poin from D \ {}. Lemma B. e.g., Goluzin [4]) If f S and w / fd), hen a 2 + w 2. In paricular, w /4 wih equaliy only for Koebe funcions. Lemma C. Barnard, Schober []) If f S c and w / fd), hen a 2 + w τ = 2 sin x cos x.327, x where x 2.86 is he unique roo of he equaion x co x = 2 x2. The resul is sharp.

3 LINEAR INVARIANCE AND INTEGRAL OPERATORS OF UNIVALENT FUNCTIONS 3 Lemma D. Sheil-Small [5], Suffridge [6]) Le f S c and ξ D be fixed. Then he funcion gz) = ξz fz) z ξ belongs o he class of sarlike funcions of order /2. From Lemmas D and A, we obain Lemma. Le f S c and ξ D. formula 7) G α, z) = Ĵα,f)z) = where g is a sarlike funcion of order /2. Then he family M α, S c ) is given by he f )) α ξ) 2 2α ) g) d, Remark. Noe how he family of he inegrals given in 4) and he minimal invarian family, which conains 4), given in 7) differ from each oher every convex funcion is sarlike of order /2 in D). Lemma 2. The order of he family M is given by he formula 8) ord M α, = sup αa 2 + α)ξ + 2 ξ ξ ). f M ξ D Proof. By 3) we have o calculae 2 F α, ) which gives us he resul. Lemma 3. If fz) = kz) = z/ z) 2, hen + 2 ) if α =, R, α 2 α + α if < α 9) ord Ĵα,k)) = 2, α + 2 α ) if 2 α, α + 2 α + α if α. Proof. Formula 8) gives ord Ĵα,k)) = sup 2α + α)ξ + ) ξ)2 ξ 2 ξ ξ which implies he resul. ξ D = sup θ 2π 2α + α)e iθ cos θ α + sup θ 2π 2α + cos θ) For compleeness, le us wrie Lemma 4. If fz) = lz) = z/ z), hen 2α + if 2α + 2, ) ord Ĵα,l)) = if 2α 2α + 2, 2α if 2α.

4 4 MICHAEL DORFF AND J. SZYNAL 3. New Resuls Now we find ord M α, S) and ord M α, S c ). Theorem. We have Proof. Le Then ord M α, S) = ordĵα,k)). ν α, f) = sup αa 2 + α)ξ + 2 ξ ξ ). ξ D ord M α, S) = sup ν α, f). f S Nex, noice ha ν α, f) α + sup αa ξ ξ ). ξ D If =, hen ν α, f) α + 2 α = ord M α, S). If, we observe ha he leas upper bound for αa ξ ξ ) is aained if ξ. To see his, suppose o he conrary ha here exiss ξ D such ha ν α, f) = αa 2 + ξ ) 2 ξ fξ ) = Re [e iγ αa 2 + ξ ) )], γ R. 2 ξ fξ ) Bu he righ hand side of his equaion is a harmonic funcion which canno aain is maximum inside D. This conradicion proves our observaion and we can wrie: sup αa 2 + ξ D 2 ξ ξ ) = lim sup ξ αa ξ ) ξ = lim sup αa 2 ξ 2 2 if α =, = α sup 2α fc) if α. Using Lemma B and he fac ha sup 2α c = c fd) we have o consider he following cases: c fd) sup w / fd) 2α w a) if 2 /α, hen by Lemma B and he inequaliy a 2 2 we have a 2 2αw = 2α + ) a 2 2α 2αw w ) a 2 + 2α w + + ) a 2 2; 2α

5 LINEAR INVARIANCE AND INTEGRAL OPERATORS OF UNIVALENT FUNCTIONS 5 b) if /α 2, hen Lemma B and w /4 yields 2α w a 2 + w + ) 2 2 w 2α α ; c) if /α, hen as above 2α w a 2 + 2α w α. This complees he proof. All bounds are sharp and obained by Koebe funcions. The case in which α = was obained in [9]. For he class of convex univalen funcions he siuaion is differen. Theorem 2. + if α =, R, ) ord M α, S c α + ) ) = α + α if α, α [ + 2α τ)] + α if 2 α, α [ τ 2 ] α + α if α 2. For he firs wo cases he bounds are sharp. Proof. As in he proof of Theorem, we have ν α, f) α + sup αa 2 + ξ D 2 ξ ξ ) if α =, = α + α sup 2α w if α. w / fd) Now he upper bound for a 2 /2α) /w will depend on he inequaliy in Lemma C and he well known fac ha w /2 for f S c. However, because hese wo inequaliies have differen exremal funcions, our bounds will be sharp only in he cases menioned in he saemen of his heorem. Therefore, we have: a) if /α, hen 2α w a 2 + 2α w + α ; b) if 2 /α, hen 2α w ) a 2 + 2α w + + ) a 2 + τ); 2α 2α c) if < /α 2, hen 2α w a 2 + w + ) 2α w τ 2 α, and he proof of Theorem 2 is complee. The following corollaries are he resuls of Theorems and 2 and he Lemmas.

6 6 MICHAEL DORFF AND J. SZYNAL Corollary. If f S c or S), hen { 2) M α, := and G α, z) = ord M α, S c ) = α + α ; } f )) α ξz) d ; f 2 2α Sc or S), ord M α, S) = α + 2 α. Corollary 2. If f S c, hen for all α [, ] and ξ D, G α, z) = f )) α ξz) 2 2α d Sc. Remark. If f S and α [, ], hen ord M α, S) = + α and herefore he ransformaion 2) decreases he order ord S = 2). This informaion implies by a heorem from Pfalzgraff [], he following sligh exension of his /4 heorem: If G α, z) M α, S) and α [, ], hen he inegral λ 2+α), λ C). Remark. From Theorems and 2 we have: ord M, S) = 3, ord M, S) = 2, ord M, 2 S c ) τ, ord M,2 S c ) = 3. G λ α, ) d is univalen for Therefore, we have examples of operaors J α, for which he minimal invarian family Ĵα, has larger, smaller, or he same order as S or S c. 4. Applicaions Now we give some applicaions of he resuls obained in he previous secion. Theorem 3. If f S c and G α, is given by 7) i.e., G α, M α, S c )), hen we have he sharp bound 3) arg G α,z) hα, ) arcsin r, z = r <, where 4) hα, ) = 2 α α 2. Proof. From 7) we have arg G α,z) α arg f z) + arggz)/z) + 2 2α arg ξz) 2 α arcsin r + arcsin r + 2 2α arcsin r = hα, ) arcsin r, by he well-known sharp roaion heorems arg f z) 2 arcsin r, f S c and arggz)/z) arcsin r for /2 sarlike funcions [5]. Theorem 4. The radius of univalence, r u, of M α, S c ) saisfies he inequaliy r u r α,, where { } π 5) r α, = min, an, hα, ) and hα, ) is given by 4).

7 LINEAR INVARIANCE AND INTEGRAL OPERATORS OF UNIVALENT FUNCTIONS 7 Corollary 3. A funcion G α, M α, S c ) is univalen in D if α, ) A, where A = { α, ) : α [, 3/2 ], [, 3 2α] } { α, ) : α [ /2, ], [ 2α, 3] }. Region A. Proof. The family M α, S c ) is a linear invarian family and herefore by a resul of Pommerenke see [2], pp ) he radius of univalence r u = r u M α, S c )) saisfies he inequaliy r u r α,, where r α, = + r 2 and r is he radius of he disk z < r in which G α, z)/z. Moreover, he value of r is deermined from he equaion r max arg G α,z) = 2π. f S c z =r Inequaliy 3) gives r = sin 2π hα,). This along wih 5) gives region A. Corollary 3 and he shape of region A are equivalen o he following inequaliy 2 hα, ) 4, which implies he univalence of M α, S c ) in he whole disk D. In fac, now he conclusion of Theorem 4 can be srenghened and made sharp. Theorem 5. The family M α, S c ) consiss of univalen close-o-convex funcions in D for α, ) A. If α, ) / A hen he radius of close-o-convexiy of M α, S c ) is he unique roo of he equaion 6) 2arcco [ r 2 h2 α, )r 2 + r 2 ) 2 ] hα, )arcco [ 2 hα, ) r2 ) h2 α, )r 2 + r 2 ) 2 ] = π The proof of Theorem 5 follows direcly from inequaliy 3) and he following lesser known resul. Lemma E [2], p. 9, Cor. 3.3). If M is a linear invarian family for which max arg f z) = 2τ arcsin r, f M z =r< hen he radius of close-o-convexiy of M is if τ 2, and if τ > 2 he radius of close-o-convexiy is he unique roo of he equaion 2arcco w 2τarcco τw) = π,

8 8 MICHAEL DORFF AND J. SZYNAL where w = r 2 4τ 2 r 2 + r 2 ) 2. Corollary 4. If α, ) A, hen he class M α, S c ) consiss of close-o-convex funcions in D and he resul is sharp by Lemma E). The exremal funcion is he half plane mapping l ɛ z) = z/ ɛz) 2, ɛ =. Remark. Because we have he inclusion J α, S c ) Ĵα,S c ), Corollary 4 remains valid for he inegral given in 4) when f S c. In paricular, we obain he following resuls. Corollary 5 []). If f S c, hen he funcion F α, z) = f )) α d is close-o-convex in D for all α [ /2, 3/2] and his resul is sharp. Corollary 6 []). If f S c, hen he funcion ) z f ) F, z) = d is close-o-convex in D for all [, 3] and his resul is sharp. Corollary 7. If f S c, hen he funcion f F α,2 2α z) = f )) α ) 2 2α ) d is close-o-convex in D for all α [ /2, 3/2] and his resul is sharp. Remark. One can observe ha our mehod of employing minimal invariance family o sudy he univalence of he inegral in 4) ogeher wih he resul of Pommerenke [2] and Campbell and Ziegler [2] gives beer resuls han he direc applicaion of he heorem of Pfalzgraff [] which is so well suied o he class S. Remark. Of course, in general, he radius of close-o-convexiy for 4) and for is minimal invarian family 6) are differen. For example, for F, 2 he radius of close-o-convexiy can be calculaed from 4) o be / 2.77, while he radius of close-o-convexiy of M, 2 S c ) can be calculaed from 6) wih h = 6 o be approximaely equal o.553. References [] R. W. Barnard and G. Schober, Möbius ransformaions of convex mappings, Complex Variables Theory Appl ) no. -3, [2] D. M. Campbell and M. R. Ziegler, The argumen of he derivaive of linear-invarian families of finie order and he radius of close-o-convexiy, Ann. Univ. Mariae-Curie-Sk lodowska, Sec A, ) [3] N. Danikas and S. Ruscheweyh, Semi-convex hulls of analyic funcions in he uni disk. Analysis 9 999) no. 4, [4] G. M. Goluzin, Geomeric heory of funcions of a complex variable, Amer. Mah. Soc., Providence, Rhode Island, 969. [5] A. W. Goodman, Univalen funcions, vol. II, Mariner Pub. Co., Inc., Tampa, Florida, 983. [6] R. R. Hall, On a conjecure of Clunie and Sheil-Small, Bull. London Mah. Soc. 2 98) no., [7] J. Krzyz and Z. Lewandowski, On he inegral of univalen funcions, Bull. Acad. Polon.Sci. Ser. Sci. Mah. Asronom. Phys. 963)

9 LINEAR INVARIANCE AND INTEGRAL OPERATORS OF UNIVALENT FUNCTIONS 9 [8] Z. Lewandowski, On a problem of M. Biernacki, Ann. Univ. Mariae Curie-Sk lodowska Sec. A 7 963) [9] Z. Lewandowski, D. V. Prokhorov, and J. Szynal, Linear-invarian order for inegral ransforms of univalen funcions, Folia Scien. Univ. Techn. Resoviensis ) [] E. P. Merkes and D. J. Wrigh, On he univalence of a cerain inegral, Proc. Amer. Mah. Soc ) 97. [] J. A. Pfalzgraff, Univalence of he inegral of f z) λ, Bull. London Mah. Soc ) no. 3, [2] C. Pommerenke, Linear-invariane Familien analyischer Funkionen I, Mah. Ann ) [3] D. V. Prokhorov and J. Szynal, On he radius of univalence for he inegral of f z) α, Ann. Univ. Mariae Curie-Sk lodowska, Sec. A ) [4] W. C. Royser, On he univalence of a cerain inegral, Michigan Mah. J ) [5] T. Sheil-Small, On convex univalen funcions, J. London Mah. Soc. 2) 969) [6] T. J. Suffridge, Some remarks on convex maps of he uni disk, Duke Mah. J ) Deparmen of Mahemaics, Brigham Young Universiy, Provo, UT 8462, USA address: mdorff@mah.byu.edu Deparmen of Applied Mahemaics, Faculy of Economics, Maria Curie-Sklodowska Universiy, 2-3 Lublin, Poland address: jsszynal@golem.umcs.lublin.pl