Research Article Radius Constants for Analytic Functions with Fixed Second Coefficient

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1 Hindawi Publishing Corporation e Sientifi World Journal Volume 24, Artile ID 89864, 6 pages Researh Artile Radius Constants for Analyti Funtions with Fixed Seond Coeffiient Mahnaz M. Nargesi, Rosihan M. Ali, 2 and V. Ravihandran 3 Department of Mathematis, College of Natural Sienes & Mathematis, California State University, 8 North State College Boulevard, Fullerton, CA , USA 2 Shool of Mathematial Sienes, Universiti Sains Malaysia, 8 Penang, Malaysia 3 Department of Mathematis, University of Delhi, Delhi 7, India Correspondene should be addressed to Rosihan M. Ali; rosihan@s.usm.my Reeived 3 February 24; Aepted 3 June 24; Published July 24 Aademi Editor: Ming-Sheng Liu Copyright 24 Mahnaz M. Nargesi et al. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. Let f(z) = z + a nz n be analyti in the unit disk with the seond oeffiient a 2 satisfying a 2 =2b, b. Sharp radius of Janowski starlikeness is obtained for funtions f whose nth oeffiient satisfies a n n+d(,d )or a n /n ( > and n 3). Other radius onstants are also obtained for these funtions, and onnetions with earlier results are made.. Introdution Let A denote the lass of analyti funtions f defined in the open unit disk D := {z C : z <},normalizedbyf() = = f (), andlets denote its sublass onsisting of univalent funtions. If f(z) = z + a nz n S,deBranges [] obtained the sharp oeffiient bound that a n n(n 2). However, the inequality a n n, n 2,isnotsuffiientforf to be univalent; for example, f(z) = z + 2z 2 is learly not a member of S. Several sublasses of S possess a similar oeffiient bound. For instane, the nth oeffiients of starlike funtions, onvex funtions in the diretion of imaginary axis, and lose-to-onvex funtions satisfy a n n (n 2) [2 4]. Other examples inlude funtions whih are onvex, starlike of order /2, and starlike with respet to symmetri points. The nth oeffiients of these funtions satisfy a n (n 2) [5 7]. The nth oeffiient of lose-to-onvex funtions with argument β satisfies a n +(n )os β [8], and the oeffiients of uniformly starlike funtions are bounded by 2/n [9], while a n /n[] for uniformly onvex funtions. Simple examples show that these bounds are not suffiient to haraterize the geometri properties of the lasses of funtions. In the sequel, we will assume that f A has the Taylor expansion of the form f(z) = z + a nz n.gavrilov[] showed that the radius of univalene for funtions f A satisfying a n n(n 2)is the real root r.64 of the equation 2( r) 3 (+r) =,andtheresultissharpfor f(z) = 2z z/( z) 2. Gavrilov also proved that the radius of univalene for funtions f A satisfying the oeffiient bound a n M(n 2)is M/( + M). The ondition a n Mlearly holds for funtions f A satisfying f(z) M, and for these funtions, Landau [2]provedthattheradius of univalene is M M 2.Infat,Yamashita[3]showed that the radius of univalene obtained by Gavrilov []isalso the radius of starlikeness for funtions f A satisfying a n n or a n M. Additionally, Yamashita [3] determined that theradiusofonvexityforfuntionsf A satisfying a n n is the real root r.9 of the equation 2( r) 4 (+4r+ r 2 )=, while the radius of onvexity for funtions f A satisfying a n Mis the real root of (M+)( r) 3 M(+r) =. () Reently, Kalaj et al. [4] obtained the radii of univalene, starlikeness, and onvexity for harmoni mappings satisfying ertain oeffiient inequalities.

2 2 The Sientifi World Journal For two analyti funtions f and g, thefuntionf is subordinate to g, denoted by f g, if there is an analyti self-map w of D with w() = satisfying f(z) = g(w(z)). If g is univalent, then f gis equivalent to f() = g() and f(d) g(d). For β R \{}, α,thelassl(α, β) onsists of funtions f A satisfying α z2 f (z) f (z) + zf (z) f (z) +( 2β)z. (2) z Denote by L (α, β) itssublassonsistingoffuntionsf A satisfying α z2 f (z) + zf (z) f (z) f (z) β (β R \ {},α ). (3) These lasses were investigated in [5 24]. For β <,thelassl(, β) is the lass of starlike funtions of order β, while, for the ase β>, the lass was studied in [25 28]. The lass ST[A, B] of Janowski starlike funtions [29] onsists of f Asatisfying the subordination zf (z) f (z) +Az +Bz ( B<A ). (4) Certain well-known sublasses of starlike funtions are speial ases of ST[A, B] for appropriate hoies of the parameters A and B. Forexample,for β <, ST(β) := ST[ 2β, ] is the familiar lass of starlike funtions of order β. DenotebyST β the lass ST β := L (, β) = ST[ β, ]. Janowski[29] obtained the sharp radius of onvexity for ST[A, B]. This paper studies the lass A b onsisting of funtions f(z) = z + a nz n, ( a 2 =2b, b ),inthediskd. The sublass of univalent funtions in A b have been studied in [3 33]. In [33], Ravihandran obtained sharp radii of starlikeness and onvexity of order α for funtions f A b satisfying a n n or a n M, n 3.Theauthoralso obtained the radius of uniform onvexity and paraboli starlikeness for funtions f A b satisfying a n n, n 3. This paper finds radius onstants for funtions f(z) = z+ a nz n A b satisfying either a n n+d(, d )or a n /n ( >, n 3). In the next setion, sharp L(α, β)- radius and ST[A, B]-radius are derived for these lasses. Several known radius onstants are shown to be speial ases of the results obtained. 2. Radius Constants A suffiient ondition for funtions f A to belong to the lass L(α, β) isgiveninthefollowinglemma. Lemma (see [24, 34]). Let β R \{}and α.iff(z) = z+ a nz n A satisfies the inequality then f L(α, β). (αn 2 + ( α) n β) a n β, (5) Making use of this lemma, the sharp L(α, β)-radius is obtained for f A b satisfying the oeffiient inequality a n n+d. Theorem 2. Let β R \{}, 6α + 3 β,andα.the L(α, β)-radius for f(z) = z + a nz n A b satisfying the oeffiient inequality a n n+d,, d, n 3,isthereal root in (, ) of the equation ((+d)( β) + β +(2α+2 β)(2 ( b) +d) r) ( r) 4 =α(+4r+r 2 )+(( α) +αd) ( r 2 (6) ) +(( α) d β)( r) 2 βd( r) 3. For β<,thisnumberisalsothel (α, β)-radius of f A b. The results are sharp. Proof. The number r is the L(α, β)-radius for f A b if and only if f(r z)/r L(α, β). Therefore, by Lemma, it is suffiient to verify the inequality (αn 2 + ( α) n β) a n rn β, (7) where r is the real root in (, ) of (6). Using the known expansions leads to r n = nr n =, (8) ( r ) 2 2r, (9) n 2 r n = +r ( r ) 3 4r, () n 3 r n = +4r +r 2 ( r ) 4 8r () (αn 2 + ( α) n β) a n rn 2 (2α + 2 β) br + (αn 2 + ( α) n β)(n + d) r n = 2 (2α + 2 β) br +α( +4r +r 2 ( r ) 4 8r ) + (( α) +αd) ( +r ( r ) 3 4r ) +(( α) d β)( ( r ) 2 2r ) βd( )

3 The Sientifi World Journal 3 = (+d) (β ) (2α + 2 β) (2 ( b) +d) r +(α(+4r +r 2 )+(( α) +αd) ( r2 ) +(( α) d β)( ) 2 βd( ) 3 ) ( ) 4 = β. (2) For β<, onsider the funtion f (z) =z 2bz 2 (n + d) z n = (+) z+2( b) z 2 At the root z=r in (, ) of (6), f satisfies where Re (α z2 f (z) f (z) z ( z) 2 dz3 z. (3) + zf (z) f (z) )= N(r ) =β, (4) D(r ) N(r )= 2( b)(2α + ) r + 2r (2α + ) ( r ) 3 + 6αr2 ( r ) 4 + 2dr 2 (3α + ) 4 α + dr3 (6α + ) ( r ) 2 + 2dr ( r ) 3, D(r )=++2( b) r ( r ) 2 dr. 2 (5) This shows that r is the sharp L(α, β)-radius for f A b.for β<,(4) shows that the rational expression N(r )/D(r ) is positive, and therefore the equality α z2 f (z) + zf (z) f (z) f (z) = β (6) holds. Thus, r is the sharp L (α, β)-radius for f A b when β<. For β>,thefuntion f (z) =z+2bz 2 + (n + d) z n = ( ) z+2(b ) z 2 + z ( z) 2 + dz3 z (7) demonstrates sharpness of the result. The derivation is similar to the ase β<and is omitted. Theorem 3. Let β R \{}and α.thel(α, β)-radius of f(z) = z + a nz n A b satisfying the oeffiient inequality a n /nfor n 3and >is the real root in (, ) of the equation [ ( β) + β +(2α+2 β)r( 2 2b)]( r)2 =α+( α) ( r) +β( r) 2 log ( r). r (8) For β<,thisnumberisalsothel (α, β)-radius of f A b. The results are sharp. Proof. By Lemma, r is the L(α, β)-radius of funtions f A b when inequality (7) holds for the real root r of (8) in (, ).Using(8)and(9)togetherwith leads to rn n = log ( r ) r r 2 (αn 2 + ( α) n β) a n rn 2 (2α + 2 β) br + (αn 2 + ( α) n β)( n )rn = 2 (2α + 2 β) br +α( ( r ) 2 2r ) + ( α) ( ) β( log ( r ) r r 2 ) =(β )+(2α+2 β)r (2b 2 ) (9) + αr + ( α) ( )r +β( ) 2 log ( r ) ( r ) 2 r = β. (2) To verify sharpness for β<, onsider the funtion f (z) =z 2bz 2 n zn = (+) z+( 2 2b)z2 +log ( z). (2)

4 4 The Sientifi World Journal At the root z=r in (, ) of (8), f satisfies Re (α z2 f (z) + zf (z) f (z) f (z) ) = ( ( 2 2b)r r (2α + ) + α ( r ) log ( r ) ) r ((+) +( 2 2b)r + log ( r ) ) =β. r (22) Thus, r is the sharp L(α, β)-radius for f A b.forβ<, the rational expression in (22) is positive, and therefore α z2 f (z) + zf (z) f (z) f (z) = β, (23) whih shows that r is the sharp L (α, β)-radius for f A b. For β >, sharpness of the result is demonstrated by the funtion f given by f (z) =z+2bz 2 + n zn = ( ) z+(2b 2 )z2 log ( z). (24) Remark 4. The results obtained above yield the following speial ases. () For α =, β =, =, d =,and b, Theorem 2 yields the radius of starlikeness obtained by Yamashita [3]. (2) For α=, =,andd=, Theorem 2 redues to Theorem 2. in [33, page3].whenα=, =,and d=m, Theorem 2 leads to Theorem 2.5 in [33, page 5]. (3) For α=, Theorem 3 yields the radius of starlikeness of order β for f A b obtained by Ravihandran [33, Theorem 2.8]. The following result of Goel and Sohi [35]willberequired in our investigation of the lass of Janowski starlike funtions. Theorem 6. Let B<A.TheST[A, B]-radius for f(z) = z + a nz n A b satisfying the oeffiient inequality a n n+d, n 3and, d, is the real root in (, ) of the equation [(A B)(+d+) (2b 2 d)(2 ( B) ( A)) r] ( r) 3 Thisradiusissharp. =( B)(+r) + (d ( B) ( A)) ( r) ( A) d( r) 2. (26) Proof. It is evident that r is the ST[A, B]-radius of f A b if and only if f(r z)/r ST[A, B]. Hene,byLemma 5, it suffies to show that (( B) n ( A)) a n rn A B ( B<A ), (27) where r is the root in (, ) of (26). From (8), (9), and (), it follows that (( B) n ( A)) a n rn 2(2 ( B) ( A)) br + (( B) n ( A)) (n + d) r n =2(2 ( B) ( A)) br +( B) ( +r ( r ) 3 4r ) + (d ( B) ( A)) ( ( r ) 2 2r ) ( A) d( r ) = (B A)(+d) + (2b 2 d) Lemma 5 (see [35]). Let B<A.Iff(z) = z + a nz n A satisfies the inequality then f ST[A, B]. (( B) n ( A)) a n A B, (25) The next result finds the sharp ST[A, B]-radius for f A b satisfying the oeffiient inequality a n n+d. (2 ( B) ( A)) r +(( B) ( + r ) + (d ( B) ( A)) ( r ) ( A) d( r ) 2 ) ( ) 3 =A B. (28)

5 The Sientifi World Journal 5 The funtion f given by (3) shows that the result is sharp. Indeed, at the point z=r where r is the root in (, ) of (26), the funtion f satisfies zf (z) f (z) =( 2( b) r + 2dr2 dr 3 + ( r ) 2 + 2r ( r ) 3 ) (++2( b) r ( r ) 2 dr ) A B zf (z) f (z) (+)(A B) +2( b) r = (A 2B) ++2( b) r /( ) 2 dr 2/() Then, (26)yields (A B) ( ( r ) 2 + 2r B ( r ) 3 dr2 (A 3B) + dr3 B ( r ) 2 ) (++2( b) r ( r ) 2 dr ) zf (z) f (z) = A B zf (z) f (z) or equivalently f ST[A, B]. 2 2., (29) ( B<A,z=r ), (3) Proof. By Lemma 5, ondition (27) assures that r is the ST[A, B]-radius of f A b where r is the real root of (3). Therefore, using (8)and(9)forf A b yields (( B) n ( A)) a n rn 2(2 ( B) ( A)) br + (( B) n ( A)) ( n )rn =2(2 ( B) ( A)) br +( B) ( ) ( A) ( log ( r ) r r 2 ) =(B A) + (2 ( B) ( A)) r (2b 2 ) + ( B) r +( A) ( r ) log ( r ) ( r )r =A B. (32) The result is sharp for the funtion f given by (2). Indeed, f satisfies zf (z) f (z) = (/2 2b) r +/( )+(log ( r )) /r (+) + (/2 2b) r +(log ( r )) /r, A B zf (z) f (z) =((+)(A B) + (A 2B) ( 2 2b)r Theorem 7. Let B<A.TheST[A, B]-radius for f(z) = z + a nz n A b satisfying the oeffiient inequality a n /n, n 3and >, is the real root in (, ) of the equation ((+)(A B) (2 ( B) ( A)) r(2b 2 )) + B + A log ( r ) r ) ((+) +( 2 2b)r + log ( r ) ), r (33) ( r) =( B) +( A)( r) Thisradiusissharp. log ( r). r (3) at the root z=r in (, ) of (3). Evidently, the funtion f satisfies (3),andhenetheresultissharp. Conflit of Interests The authors delare that there is no onflit of interests regarding the publiation of this paper.

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