CONVEX FUNCTIONS AND THE ROLLING CIRCLE CRITERION

Given 0≤R1≤R2≤∞, CVG(R1,R2) denotes the class of normalized convex functions f in the unit disc U, for which ∂f(U) satisfies a Blaschke Rolling Circles Criterion with radii R1 and R2. Necessary and sufficient conditions for R1=R2, growth and distortion theorems for CVG(R1,R2) and rotation theorem for the class of convex functions of bounded type, are found.

Let S be the class of functions f(z) which are analytic and univalent in the unit disc U z: z < 1 and have the normalization f(O) 0 f' (0)-I.For f S and r (O,l),the radius of curvature, p(z) of the curve f(Izl r) at the point f(z), is given by [6], z' (z) p(z) zf" (z) Re(l + 7 (z)) i8 where z re Goodman [2] introduced the class CV(R I,R2) of functions f(z) having p(z) restricted as Izl tends to i. Thus, let DEFINITION I. Let R] and R 2 be fixed in [0,=}.A function f S_ , is said to, be in the class CV(R1,R2) if R 1 -R, and R R 2 where R, and R are as in (i).For 0 < R I R2< , a function f CV(RI,R2 is called a convex _function of bounde_d_type. , A function, f(z) is said to be in -Q(RI,R2) if, RI= R, and R2=R where R, and R are as in (1).
For functions f(z) in the class CV(RI,R2) Goodman [2] obtained (i) the first approximation for the moduli of the Taylor coefficients, (ii) covering theorem and (iii) bounds for d, where d is the distance of af(U) from the origin, in terms of R 1 and R 2.
Goodman [3], Wirths [8] and Mejia and Minda [4] extended this study by finding certain other interesting properties of functions in the class CV(RI,R2).
Styer and Wright [7] introduced the following class of functions based on Blaschke's Rolling Circles Criterion: R and CVG(RI,R2) be the class of functions f(z) in with the property that for each n af(U) there are open discs D l(n) and D2(n of radius R 1 and R2, respectively, such that, n e aDl(n N aD2(n) and D I() f(U) -D 2(D).
If R 1 0 or R 2 , DI(W) and D2(n are to be interpreted as the empty set and an open half-plane, respectively.It follows that [7]   CV(RI,R2) _ CVG(RI,R2) K CV where, CV is the subclass of functions f(z) in the class S, for which f(U) is convex.
However, for R 1 > 0, whether CVG(RI,R2) CV(RI,R2) still holds, remains an open problem.The difficulty to settle this problem lies in the fact that, for f CVG(RI,R2), R 1 > 0, the radius of curvature p(z) of the curve f(Izl r) at the point f(z) may not be a continuous function on U z Izl -1 ), (see [7]).
Let g(z) be analytic and univalent in U. A function f(z) analytic in U, is said to be subordinate to g(z) in U (f(z) g(z)) if f(O) g(O) and f(U) g(U).For a function f(z) in S, the unit exterior normal to the curve f(Izl zf'(z)/Izf' (z) where r (0,i).Styer and Wright [7] found that a normalized univalent function f CVG(RI,R2), if and only if, f CV, and for every and, in the case R 1 > 0, (3) D(f() Rln(< ,RI) f(U).
where D(a,R) is the open disc of radius R cenetred at a.For a function f(z) in the class CV(RI, R2) Goodman [2] obtained bounds for d and d* where the right hand side inequality in (4) and the left hand side .v,.-''-'2-,-(5/' ,2f, f i -+ f" rurther' Styer and Wright [7] observed that inequalities (4) and (6) continue to hold for the class CVG(RI,R2).The method of proof of inequality (5) in [2] shows that this inequality also holds for the class CVG(RI,R2) and is sharp.These inequalities are necessary conditions properties involving d or bound on larg f' (z) for functions f(z) in the class CVG(RI,R2) have not been investigated so far.
Section 2 is aimed at the determination of necessary and sufficient conditions for R 1 to be equal to R2, if the function f(z) is in the class CVG(RI,R2).In this section analogues of conditions , (4) and (5) involving d in place of d, for the functions in the class CVG(RI,R2) are also found.Section 3 consists of theorems on the growth of If(z) for functions f(z) in the class CVG(RI,R2).
Finally, Section 4 consists of a distortion theorem for the class CVG(R1,R2) and a rotation theorem for the class CVG(RI,R2). 2.

PRELIMINARIES.
For a function f e CVG(R 1 ,R2) we first find some relations between the smallest and the largest distances of the image curve f(U) from the origin.We first prove the following lemma (i) Follows by ( 4) V. SRINIVAS, O. P. JUNEJA AND G. P. KAPOOR disc of radius R If the center of the disc is at The function FR(Z of Lemma 1 with R R 2 (denoted as FR2(Z in the sequel) was first used by Goodman [2] as an extremal function for a number of problems concerning CV(RI,R2).
The inequalities are sharp for the function is increasing in x if 1 s x < and is decreasing in x if 1/2 s x <i.Thus inequality (7)  follows from inequalities (6) and (5).
The function FR2(Z of Lemma l(iii) is in the class CVG(RI,R2) , with d i/(i 41 I/R 2 and gives sharpness for inequality (7).
REMARK.For f CVG(RI,R2), inequality (7) sometimes gives a better lower bound on R 2 than that of inequality (5).
RE4ARKS, (i) For f CVG(RI,R2) with R 1 z I, it is easily seen that inequality (8) sometimes gives better upper bound for R than that given by inequality (5).In fact, (d* 2/ * 1 (2d -i) < d2/(2d-l), if and only if, C(RI,R2) with R 1 < i, inequality (8) is not sharp because R 1 < 1 (d* 2/ (24 -i) . GROWTH OF IF(Z) I. His proof shows that inequality (ii) continues to hold for the class CVG(R I,R2) also.However, analogues of inequalities (II) and ( 12) involving d sup I<I, are not known.In this af(u) section these analogues are derived.
for Izl r [r ,I) where r 2R2(R2-d)/(2R2(R2-d) + d 2) and the inequality is sharp.In this section an analogous inequality for the functions in the class CVG(RI,R2) is found wherein the number r is independent of d.
In the following proposition, an analogue of inequality (ii) , involving d in place of d is found.In Theorem i, an improvement of this proposition will be obtained.P0POSII0N 4. If f CVG(RI,R2) with R 2 < -, then (13) f(z r(R 2 + IR2 d* in the disc z r s i.The inequality is sharp for R 1 R 2.

P00F.
From the definition of d we have that , in the disc zl r s i.The triangle inequality and Schwarz lemma together with the above inequality completes the proof of (13).
PROOF.The inequality in the corollary is straightforward in view of inequality (13).
d < (iii) An analogue of inequality (13) involving R 1 can also be found.Thus, if f CVG(R1,R2) with R 2 < , then in the disc zl r s i.The above inequality is sharp for R 1 R 2.
Next, a growth theorem is derived for the class CVG(RI,R2) with the help of the following lemma: LEMMA 2 [5].
If F (z) is in CV and f (z) is convex and univalent in U, then f(z) F(z) in U implies that Iz:l - IF(z) in the disc I.I < R, where _R 0.543 is the least positive root of arc sin x + 2 arc tan x -2 where Izl r, the left hand side inequality holds in the disc zl < _R, B is as in Lemma 2 and the right hand side inequality holds in the disc zl s i.Both the inequalities are sharp.in the disc zl < R where _R is as in Lemma 2. This gives the left hand side inequality of (14).

R[MARKS.
(i) For f CVG(RI,R2) with R 2 < and r i the upper bound of l.f(z) in inequality ( 14) is larger than that given by inequality (13).For the function FR2(Z of Lemma l(iii), both the bounds are equal.
For r < i, the upper bound given by inequality ( 14) is better than that given by inequality (13).
(ii) From the proof of Theorem l,it can be observed that inequality (14) with d* replaced by d everywhere, continues to remain true and sharp; i.e., if f CVG(RI,R2

R2(l_r)+rd
where Izl r, the left-hand side inequality holds in the disc Izl < R, R is as in Lemma i, and the right hand side inequality holds in the disc Izli.The same function FR2(Z) of Lemma l(iii) gives the sharpness in this inequality also.
(iii) Let Q(r,R2,x) x(2R2-x)/(R2-1R2-xlr) It can be seen that for r [r ,i), the function Q(r,R2,x) is decreasing in x for x -R 2 and hence the upper bound of If(z) in , inequality ( 14) is better than that in inequality (16) for R 2 z d where r 2 R2-R2/(2R2-I) (iv) Let p(r,Rl,X xl2Rl-Xl/(Rl+IRl-Xlr).It can be seen that for r [0,R), R is as in Lemma 2, the function P(r,Rl,X) is decreasing in x for x [RI,2RI] and hence the lower bound of If(z) in inequality ( 16) is better than that in inequality (14) for , R 1d -* d -2RI; the last inequality does hold for the function p3(z) z+az 3 CVG((I+3a)2/(I+9a),R2), where 0 s a s 1/15.f(z) has R 1 R 2. For f CVG(RI,R2), the upper bound of If(z) in inequality (14) (or ( 16)) is dependent on d* (or d).The following theorem gives an upper bound of If(z) that is independent of both d and d*.
The function FR2 z of Lemma 1 iii gives sharpness in inequality (17) for z r.

REARKS. (i)
For f e CVG (R2,R2) the upper bound of f (z) in inequality (17) is better than that in inequality (13).Indeed, for the function Q (r,R2) rR2/(R 2 r R2-R 2) we have C--q(RI,R2) and equality holds in (17), then as in Remark (v) following the proof of Theorem I, we obtain that RI= R 2.
In the following result an upper bound on If(z) involving both R 1 and R 2 is obtained.RI-R1 in rQ(r,R2,d) and obtain the assertion from inequality (16).
The function FR2(Z of Lemma l(iii) gives sharpness in inequality (18) for z r. (ii) For f C-V(RI,R2) with 1 -< R 1 < R 2 < m, strict inequality holds in (18) for, when equality holds, it can be seen as in Remark (v) following the proof of Theorem i, that R 1 R2, a contradiction.DISTORTION AND ROTATION THEOREMS.
From the proof of inequality (19), we observe that inequality (19) continues to hold for the class CVG(RI,R2).However, an analogue of , inequality (19) in terms of d sup II is not known.In this e 8f (U) section a result in this direction is found for the class CVG(RI,R2).
Finally, in this section, a rotation theorem is derived for the class CV(RI,R2).Its validity for the class CVG(RI,R2) remains open for investigation.
distances of the nearmost and the farthermost points on af(U) from the origin.Thus he proved that and _] 2 ), for some real (iii) f(z) e iu FR(Z e-iU), where FR(Z (d*) 2/ (2d*-l) > d2/(2d-l), if and only if d(2d-l) < d* There does exist a function in the class CVG(R 1 R 2) satisfying d/(2d-l) < d* consider for example, f(z) 21og(l-z/2) -I CV(I, 2/-3) PROPOSITION 2. zf f CVG(RI,R2) with R 1 -I, then * By making a suitable rotation of f(z}, we may i8 assume that f(e o) -d* Then the unit exterior normal to af(U) at ie ie f(e o) is n(e o)= -i.And, by the containment relation (3), we have , D(RI-d RI) f(U) For f CV(RI,R2),Goodman ([2],[3]) found that l-r) + rd in the disc zl r -1 where d inf II-Both the inequalities f(u) are sharp.
P00[.By making a suitable rotation of f(z) we may obtain that ie f(e o) -d* sup II, for some 8 real.We have n(e o)= -i.af(u) o Now, the by containment relation (2), we get , f(U) = D (R2-d ,R2) or I-Az where B d*(2R2-d* )/R 2 and A (R2-d*)/R 2.The inverse of the function g(z)Bz/(l-Az) is h(z) z/(Az +B) and the function n(z) (hof) (z) satisfies the conditions of Schwarz lemma.So, l(z) r(IAf(z)+ B)    in the disc zl r s I.This implies that rlA By substituting the values of A and B in this, the right hand side inequality of (14) is obtained.Now, to prove the left hand side inequality in (14), we apply the containment relation (3) and obtain , Q(RI,R2) with R 1 < R2, strict inequality holds in the right hand side of the inequality (14), because, when equality holds, inequality (15) gives that f(z) Cz/(l-Dz) where C e i# d*(2R2-d*)/R 2 and D e i# d*(R2-d*)/R 2 # real, so that The inequality is sharp for R 1 R 2.PROOF.set Q(r,R2,d d(2R2-d)/(R2(l-r + rd).

,
RMARK$.(i) The number r defined in Theorem 2 is larger than ** r defined in Theorem 3.Both are equal, if and only if, R 1 R 2.