DISTORTION MAXIMUM RESULTS FOR HYPERBOLICALLY CONVEX FUNCTIONS by GERARD L. ORNAS, JR., B.S., B.S., M.S. A DISSERTATION IN MATHEMATICS Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY
Approved
August, 2003
ACKNOWLEDGEMENTS Let e < 0. I would like to thank everyone who helped me in the preparation of this dissertation. In particular: the co-chairs of my dissertation committee, Professors Roger W. Barnard and Kent Pearce for their guidance, help, and patience both on this paper and in my professional development. Also I would like to thank the graduate advisor who brought me here and has been among the nicest and most helpful people I have met. Dr. Harold R. Bennett. Next, I would like to thank my friends for helping me survive, in class, with my teaching, and just life in general, to name just a few: Shabnam Behesthi, Kimberly Drews, Jake Kesinger, Cindy Martin, Clint Richardson, and Ed and Kimberly Swim. (If I've forgotten anybody, don't worry, no one is going to read this anyway!) Finally, I would like to thank my family for all their help, patience, and for the past fifteen years of college. With your help I finally finished and got a job. I would also like to thank the Graduate School for my Summer 2002 dissertation Research Grant.
n
CONTENTS ACKNOWLEDGEMENTS LIST OF FIGURES I. II. INTRODUCTION VARIATIONS ii iv 1 7 14 22 23
III. VARIATIONAL TECHNIQUES AND RESULTS IV. CONCLUDING REMARKS BIBLIOGRAPHY
m
LIST OF FIGURES
1.1 2.1 2.2 2.3 2.4 2.5 3.1 3.2
The image of Variation at an Internal Angle Construction of^4B~, C, and \'ariations at a Cusp . . . . Mappings for Variations of a Cusp . . . .
AQBO
fc^
2 8 9 11 12 13 16 20
Side Adding Variation (Right Angle Illustrated) Varied Map and Preimages of Corners Image of / , for |z| = .70
IV
CHAPTER I INTRODUCTION
A classical problem in Geometric Function Theory is to maximize the value of a given functional over a given class of analytic functions. The most famous example is the Bieberbach conjecture of maximizing the n*'' coefficient of the power series expansion of a Schlicht function. Another common problem is the distortion problem: Given a class of functions P, and a point 2 e D, maximize \f'{z)\ over P. Frequently occurring classes for these problems are the Schlicht functions, convex functions, and starlike functions, with various normalization conditions on each. A class of functions which has received much attention in the past few years is the class with which we will deal here, the hyperbolically convex functions. The study of hyperboUcally convex functions goes back to at least the 1950s. Their early applications were in the areas of Riemannian surfaces and Fuchsian groups. In 1987, Minda [7] gave one of the earliest papers using hyperbolic convexity in the context of Geometric Function Theory. In 1994, Ma and Minda [3] gave
their first general exposition on hyperbolically convex functions. In this paper, they gave an analytic characterization of hyperbolically convex functions and results on growth problems. They followed this up in 2000 [4] with another paper in which they continued their work and gave growth bounds for, among other operators, the Schwarzian derivative. In 2000, Mejia and Pommerenke [5] also began working on hyperbolically convex functions. They conjectured that the Schwarzian derivative is maximized by the hyperbolic strip map. This was shown recently by Barnard, Cole, Pearce, and Williams [2]. They also gave a conjecture on the distortion problem. We have just discovered that this was solved in a recent paper by Mejia, Pommerenke, and Vasileyev [6]. In this they showed that the function maximizing |/'(2)| for a fixed z is ka which maps the unit disk onto a disk with a lune bounded by a hyperbolic line removed:
k^{z) ^ c lO l-z
where 0 < a < 1.
2az
+ s/{l-z)^
+ 4a^z
(1.1)
Figure 1.1: The image of ka .
We now provide the necessary terminology from hyperbolic geometry. We begin by noting that we can view the hyperbolic plane as the unit disk D = { 2 e C : | 2 | < l } with the metric X{z)\dz\ = r^. As points move towards the boundary T =
I" ^
(z : j^l = 1), the distance from the origin becomes unbounded. Using this metric the geodesies (i.e., the shortest paths connecting two distinct points) are arcs of Euclidean circles which meet T orthogonally. We also note that there is a unique geodesic connecting any two distinct points. (To wit: three nonlinear points uniquely determine a circle. While we do not use three points to determine a geodesic, the requirement that the geodesic meet T orthogonally eliminates the third degree of freedom.) This view of hyperbolic geometry, due to Poincare, is not the only one, but is the one most easily suited to our needs her(
In Euclidean space, \\v say a set, A, is convex if the line segment, /, between any two points a^be A (i.e., the geodesic botween a and b) also lies entirely in A. Using
this terminology, it is easy to generalize the notion of convexity to the hyperbolic disk. A set /I C D is hyperboUcally convex provided the hyperbolic geodesic joining arbitrary a,b e A (either a circular arc or a segment of a diameter) lies entirely in A. A set may be hyperbolically convex whether or not it is convex in the Euclidean sense, and vice versa. For example, a Euclidean disk is convex in both senses. On the other hand, if we take the closed Euclidean square with vertices 7 + U, 7 - T^ T + \i
1 4 4 ' 4 4 ' 4 4
^^^ 4 - 'ih we can see that it is convex as a square is convex in Euclidean space. However, the hyperbolic geodesic joining vi = \ + \i and V2 = I - \i lies outside the square. Hence, this square is not hyperbolically convex. Similarly we consider the hyperboUc square with vertices vi = \ + ^i,, V2 = ^ + \i, V3 = ^ + =j-i, and V4 = \ + =^i. The square is hyperbolically convex, yet the Euclidean segment between any two adjacent vertices lies outside the square. So, we have a set which is hyperbolically convex, but not convex in the Euclidean sense. We now define a function, / , to be hyperbolically convex provided: / : D ) D is univalent and analytic and /(D) is a hyperbolically convex subset of D. The set of all hyperbolically convex functions, / , mapping the origin to the origin we will denote by H. Clearly, the identity function I{z) = z and its contractions, c{z) az with |Q;| < 1 are hyperbolically convex and members of H. For the existence of other hyperbolically convex functions, we rely on one of the central theorems of complex function theory, the Riemann Mapping Theorem. The Riemann Mapping Theorem says that given any simply connected proper open subset G of C, there exists an analytic map / sending D univalently onto G such that for some specified a e G, /(O) = a and /'(O) = Ci > 0. With this normalization, we can write / : D (? as >
f{z) =a + aiz + a2z'^+ a3Z^-i As a result, we have a bijective co:
(1.2)
ence between hyperbolically convex
functions in H and hyperbolically convex domains containing the origin. As a benefit of this, we can uniquely discuss a function, / , not only as rules defined by a huge morass of mathematical notation, but also as the image of the unit disk under / . The second way is much easier to visualize and discuss. We define a hyperbolic polygon to be a simply connected region bounded by a Jordan curve consisting of a finite collection of hyperbolic segments and arcs of the unit circle. The hyperbolic segments internal to D we will refer to as proper sides. The class of all functions mapping B onto hyperbolic polygons normalized as above we denote Hpoiy. The subclasses of Hpoiy of functions mapping D onto polygons with at most n sides are denoted i?. Note that HQ c Hi C H2 C . Further we have Hn S Hpoiy C H for all n e N. We next note that any hyperbolically convex region can be approximated with arbitrary precision by a hyperbolically convex polygon. In other words, Hpoiy is dense in the space of hyperbolically convex functions. Furthermore, the class of hyperbolically convex functions, with the identically zero function added is a compact family of functions. Thus, for a real-valued continuous functional L on H, in order to maximize Lf for all hyperbolically convex functions we need only consider those which are the limits of sequences of functions mapping B onto hyperbolically convex polygons. Our main result, Theorem 1.0.1, states that for any of our functionals, an extremal function lies in Hi, and thus must be a rotation of ka for some Q G (0,1), a rotation of the identity function I{z) z, or the zero function Z{z) = 0. A classical result of complex analysis is that normalized convex univalent functions can be analytically characterized. That is: a function / : D -^ C, /(O) = 0 = l - / ' ( 0 ) is convex if and only if
^i^^iw) ^ ' ^ ^^
so on can also be determined based on ' is of certain functionals.
^^ ^^ -
Other geometric properties of functions, for instance whether a function maps onto a domain starhke with respect to t.ViP nrimn, whether it is close-to-convex, and
Analogous characterizations have been found for hyperbolically convex functions [3] and [4]. For instance, for a function / analytic and locally univalent in D, we have / is hyperbolically convex if and only if
, , , ^f"(z)
2:/(;)/'(z)\
This characterization shows one of the main frustrations with hyperbolic convexity. The first two summands of the operator are exactly the same as the functional used in equation 1.3. Unfortunately, this innocuous looking third summand contains a modulus and a conjugation. These have the effect of rendering the function that results from applying a function to this operator non-analytic. This further leads to many of the tools of classical complex analysis being not applicable to problems dealing with hyperbolically convex functions. Our main result, proved in Chapter III, uses a method of Barnard, based on Julias formula for Hadamard variations, to show the function maximizing a wide class of functionals maps D onto a hyperbolic polygon with at most one side. The functional L^f \f'{z)\ is one of the functionals in this class and we therefore show the distortion problem as a corollary, using an entirely different method as Mejia, Pommerenke, and Vasileyev. Theorem 1.0.1. Let $ be real entire. Let z eB and f e H be extremal for L/(z)=!R(t(log|;|^)) such that 1. 4 ' (log ^ ) R \ {0} (1.5)
then f maps the unit disk onto a hyperl "
' igon with at most one proper side.
We make note here that in maximizing Lf{z) for any z eBwe z G (0, 1). This is due to the rotational invariance of our class.
need only consider
Our proof of Theorem 1.0.1 is based on the Julia variational formula. This is best viewed geometrically. Let f H map B -^ G C B such that dG is piecewise analytic with right and left tangents at all points. For w e dG, let n{w) be the outward unit normal where it exists and the zero vector where it does not. We define a function, piecewise differentiable, (f):dG -^ R with (t){wj) = 0 where {wj} is the collection of points at which dG is not analytic. We can define a new curve dC] pointwise by letting uT, = w + t(j){w)n{w). By choosing e sufficiently small, dG] is a Jordan curve. We now define G, to be the region bounded by dG] (with the obvious abuse of notation overlooked). We define /^ to be the Riemann map sending D onto G^ such that /,(0) = 0. Julia's result (which was really a generalization of Hadamard's work with Green's functions) was that we can write /e as a variation of our original / . In particular: 7f \ i-/ \ , ^^f'i^) f C + z <j)iw)n{w) , , , . .
[w = /(e'^), 0 < ^ < 27r), where o{e) is analytic for all z eD. A problem encountered in using the method of Julia variations with hyperbolically convex functions is the difficulties in finding Julia variations we can perform on the sides of the approximating polygons that leave the varied functions in the original class. We discuss in Chapter II several variations which will allow us to both keep the varied polygons in the original class (with one notable exception) and apply the variational techniques developed in Chapters II and III to prove Theorem 1.0.1.
CHAPTER II VARIATIONS
Now that we have our basic variational method, we will describe the actual variations we will perform. We will describe two basic types of variations. One of these will preserve the number of sides in the varied polygon. The other will increase it by one. For each type of variation, there are three cases with which we will need to deal, depending on the angles at the ends of the sides being varied. The first case is when a single side meets the boundary of the unit circle at an angle of 7r/2. The next case is the one in which two sides meet on the interior of the disk at an angle lying beytween 0 and TT. Finally, we deal with the case in which the two sides meet on the boundary of the disk at a zero angle. This case must be subdivided into two variations, one in which the side is pushed out thus turning the cusp into two right angles and one in which the meeting of the two sides is moved into the disk and the angle is increased to a positive angle. We will first explain the class preserving variations. We will use these in Chapter III to reduce the number of sides in the extremal domain to at most two. After the class preserving variations, we will explain the variations which increase the number of sides. These we will use, in the manner described by Barnard, Cole, Pearce, and Williams [2], to reduce the possible extremal domains from polygons with at most two sides to those having at most one side. The analysis of the first two cases is similar, so we will discuss those concurrently. We will illustrate by varying a side meeting the boundary of the disk on one side with an angle of 7r/2 and meeting internally another side with an angle of 6 with 0 < ^ < T T on the other (although the analysis works identically with any permutation of the two sorts of corners). See Figure 2.1 below. We consider side AB of our hyperbolically convex polygon Q. We label the point on the continuation of AB to the point G on T, allowing for the possibUlity that B = C. To perform our variation we w
7
the midpoint of AC and call it M.
Figure 2.1: Variation at an Internal Angle
Our variation will consist of moving M radially by a fixed small distance e(j){M) for constant (/)(A/). This 4){M) is chosen sufficiently small to assure that the varied polygon retains the same number of sides as the original. This will give us the new point M' - M + e(j){M)n{M). Having defined the variation at M, we now define the variation 4){w) for all other w on AB. For a given e we will define a new curve AQBQ which is the arc of the unique hyperbolic line through M' having M' as its midpoint and connecting A' to B' the resulting endpoints on dfl^ in the interior variation or the necessary extension of the original connecting sides of dQ in the exterior case. We then define the variation 4>{w, c) to be the distance to the point on ^o^o which is on the line extended along the normal n{w). Lemma 2.0.1. Fore small, expand the now two-variable variation (j){w,e) as a power series about e = 0:
^M
yjith ^ ^ jL 0.
=' ^ e
+ oie)
(2.1)
Proof: Without loss of generality we will consider only the case when e < 0. To simplify constructions and descriptions we - " " -^^so assume that M and M' are both real and negative. We will start by consi( 8 le circle C" in the plane concentric
with our original geodesic through the point M'. We will define 4>{w) to be the radial distance from w to C" Note that clearly we have
^{w) = e0(M).
(2.2)
Our strategy is to show that for each w G AB, we have that (i){w) > ^{w). Since (f){w) is sufficiently smooth, we may expand it in a power series. Suppose that Z' = 0- Then, on expanding as a function of e we get that ^{w,e) - o(e).
So if we divide 4){w, e) by ^{w) and take the limit as e goes to zero from below, we get zero, by the definition of o(e). However, as we will show (piw) > ^{w), the quotient must be greater than one for every value of e. Thus, the hmit, if it exists at all, must be greater than or equal to one. Hence, we will have a contradiction. The assumption that ^ ^ ^ ^ = o must fail and we have our result. Showing that ^{w) < (f){w) comes from a simple geometric construction. We show that except at M', the curve
AOBQ
will lie inside of C'. Since AB lies outside of C',
AQBQ
the distance from w to C' is less than the distance to
and we are done.
Figure 2.2: Construct^
^ ^B, C', and
AQBQ
The construction (see Figure 2.2) will illustrate this. Note first that A^o
and C'
both go through the point M'. Also observe that C' and AB have the same center, m. The circle, C", containing
AQBO
is normal to the unit circle at
CQ.
The tangent fine
/' to T at Co is therefore a radius of C". Note that since M' > M, we have that the center, m' of
AQBQ
lies to the left of the center of C and hence has a greater radius.
As the two circles are tangent at M', we have that the circle, C, with the smaller radius lies entirely inside the disk bounded by A^oThis gives us that A^Q lies
inside of the hyperbolic polygon whose internal boundardy is the arc of C internal to D and are we done. With this, we can now write our variation at w as
w' = w+
^^ ' 'e + o{e).
(2.3)
We can then absorb the o(e) term into the error term in the Julia Variation formula. Although the variations necessary to produce domains that are in the original class are not always strictly normal, it was shown by Barnard and Lewis [1], that the error introduced for small e is of order o(e) and thus may also be absorbed into the o(e) term in the variational formula. As the previous analysis dealt with both the first two cases, we are left only with the case in which the two sides meet at a cusp. This in turn will be dealt with in two steps. In the first case, we take e > 0 and move the arc of the circle outwards. The second case, of course, is that we take e < 0 and move the arc of the circle towards the middle of D (see figure 2.3). In the first case, we are actually for any e > 0 removing the cusp and turning it into two separate right angles at A and Ai. Note that this does not increase the number of new sides, as the new "side" lies on T and thus does not count as a proper side of the polygon. Since this variation can be done normally without moving the vertex at the cusp, the previous argum""*" '^"^M. In the second case, we will pull th 10 ightly into the disk. The arguments
Figure 2.3: Variations at a Cusp
of Barnard and Lewis for controlling the error rates need to be modified at cusps. Hence, we must use a slightly different variation. We will, geometrically, use the same variation for moving the side into the disk. It is in the analysis that we will do something new. Consider the inverse image of the perturbed domain under our original map / . This is, for small e, a simply connected region bounded by the preimage under / of the perturbed side in the disk, D and the remainder of the disk (see Figure 2.4). When e is small, we consider this inverse image as a (Julia) variation of the identity map I{z) = z. Thus, we have:
hiz) = I{z) + ezl'iz) j P(C, z)d^ + o{e) = z + ez-l [ P(C, z)d^ + 0(e).
(2.4) (2.5)
We now compose 7^ with / to get our variation on / :
11
Figure 2.4: Mappings for Variations of a Cusp
/e:=/o/,(z) = f(z + ez l'p{C,z)d^ + o{e)]
(2.6) (2.7)
Next as both I^ and / are sufficiently smooth (i.e., at least C^), we expand /^ to get
A = / . ( ^ ) U + e / ' ( ^ ) ^ ^ U + 0(6)
= f{z) + ef'{z)zl P{C,z)d^ + o{e)
(2.8) (2.9)
But this is just the standard Julia variational formula. Thus, we have the error bound derived by a modification of the Barnard and Lewis argument. And thus we have a valid application of the Julia Variation Formula for all of our class preserving variations of various angles in our polygons. We end this section with a final vari'^^"'^" ""i can apply with all three types of intersection. We will add a new small 12 3ur polygon "cutting off" a vertex
;o- This variation, unlike those previously described, will not preserve the class // but will leave the varied function in H^^i. We choose a point zi on the side of the polygon we are varying, some fixed small (Euclidean) distance CQ from ZQ. Then, choose a point cj on either the next side (if the vertex occured at a cusp or in the interior of D) or along the arc of T (if the vertex was a right angle on the boundary). Choose C some small distance e from zo along the new side. Finally join zi with 23 o with a hyperbolic geodesic (see figure 2.5). The variation will "pivot"the new side 5Tc2 about zi into the polygonal domain.
Figure 2.5: Side Adding Variation (Right Angle Illustrated)
The analysis of the error for these variations will follow very much the same path as for the previous cases. For a vertex at a right angle or an interior angle, we can use the arguments for the corresponding variations moving the entire sides. For the case of a vertex at a cusp the same argument pulling the image back, varying the identity and composing with / will proceed as with the cusp in the previous case as well. Having defined all the variations we need, we now proceed to the actual variational arguments and the proof of Theorem 1.0.1.
13
CHAPTER III \ARIATIONAL TECHNIQUES AND RESULTS
We begin our exposition of the variational analysis recalling the variational techniques just described give our perturbed functions f,{z) as functions of two variables, t and c. The basic technique with which we will establish Theorem 1.0.1 is to fix a value for z and treat Lf,{z) as a function of c. With this in mind, we will differentiate Lf^{z) with respect to e. We then expand Lf,{z) = L/,(^)|.=o + ^ ^ ^ | . = o 6 + o{e) By the definition of o(e) we have that lim (3.1)
= 0. Hence, for a sufficiently small
value of e, we have that the linear term in e dominates the o(e) term. This in turn means that if the linear coefficient of e is non-zero we can choose an e small enough that
^ ^ ' | e = o e + o()/0.
(3.2)
If the coefficient is positive, we then have a perturbed function which gives a value for L which is greater than the value given by / . Hence, / cannot possibly be extremal. Similarly, if the coeffiecient is negative, by choosing a small e < 0 we can again find a larger value for L, showing again that / is not extremal. We now refer back to equation 1.6 and write the varied function differentiated with respect to z as:
df.jz) dz
f'{z) | l + e I [{zf'iz))' P(C, z) + (zf'iz)) P'(C, z)] d * | + o{e)
(3.3)
where, d^ > 0, P(C, z) = ^ i ^ , and /;(0) = /'(O) (l + e / d*) -f o(e). Some simplification gives: 14
df.jz) dz
= /'(^)
With this expression, we compute:
^-m^wm)
(0)
2(2
(C - zY^
d^ } + o(e).
(3.4)
L/.(2) = 3?(^$(^log(^^
(3.5)
^<.(.ogm.iog(---/';^^ff.;y^(-)))).
geometric series for sufficiently small values of e. This gives
,3.a)
We continue by considering the denominator as i+,j^^^o(e) ^^^ expanding it as a
Lf{z),
= $R (^$ ( l o g | J M + log ( l + 6 y
[{znz)P{C,
Z))' - 1] d * ) + 0(6)) ) .
(3.7)
Massaging this equation slightly further, we can expand the second logarithm as a series in e for e small enough. Expanding the logarithm gives:
Lf,{z) = ? (<!. (log j^^+^j R
[(^'/'(~i^(C, z))' - 1] dJ + o{e)^^ .
(3.8)
Next, with this representation of Lf^{z) we now differentiate Lfi{z) as a function of e. Using our standard variational argument, we set e to zero and get our well prepared result:
dLf,[z)^ de
e=0
(3.9)
-A<^-9^)I[
1+
nz)\ fc+z
f'{z)J\C-zJ
2Cz -1 iC-z) (3.10) (3.11)
^K^'O-^iD/^-'^''
15
r r^s
(.
Zf"{z)\
fC + A
2(2
where 7 ^ ( 0 = I 1 +
f,\
) ( T 3 7 J - .^ _
.^ - 1- By hypothesis we have that
the first term, $ ' flogyr^^)), is real and nonzero. So we can pass the U operator through to the integral and through the integral as 0!^ is a real measure. Thus, for the derivative to be zero, we must have /3R(/2(C)) to be zero. As d^ is real valued, we have a real-valued integrand and a real-valued measure. (Recall we are integrating around the unit circle, which in turn maps onto the boundary of our hyperbolic polygon.) We now observe that, as stated in Chapter I, we can vary one side of the polygon at a time. As (/>(() = 0, for ( outside of the arc (e*^"", e*^"+i) mapping to the varied side (see Figure 3.1), our integral is reduced to:
f(D) f-(D)
Varied Side
Figure 3.1: Varied Map and Preimages of Corners
(3.12) J i(e'i,e'<'2) '(e'i,e'2) As everything is continuous and real valued, we can apply the mean value theorem for integrals to get
[
'(eii,e'2)
5R(/.(C))rf^(e'"
"^'h{m))\e=e'
fd^
(3.13)
16
where ^^ < ^- < 62. Note that as ^1 ^ 62, we have i(,.,^,..,) d^ > 0. Thus, the only way our integral can be zero is for ^ (/.(C(^))) \9=e' to be zero. With this in mind we will now examine the kernel I, of our integral. If we denote
(
zf"iz\ \
1 + "jrpr
] ^'^^ W6 combine fractions, we obtain
^HU
-i(C^ - .^) + 2C2 - (C - 2)^ ((^_ ,)2
(3-14)
Note at this point, by hypothesis, we have that A is real. Next we multiply through by 1 = I - I to get:
^..^
h{Q)
^(ICI^-(C^)^) + 2|CPC2-(|CP-C2)^
(icp - C2)2 (^-^^^
Then, we set tw = (2 to get:
-. . A{\-w^) + 2w-{l-w)'' m - ^ ^-^^^ '-.
Continuing we substitute w re'^into / and get
, , (3.16)
yl (1 - r2 (e^)') + 2re'^ - (l - re'^)' (1 - re')^ (3.17)
We then expand the above rational function, multiply through by the conjugate of the denominator, make the substitution re*^ = r (cos(^) -I- isin(^)), and collect the real parts. What is left in the numerator when all this is done is:
R (cos(^)) = - 4 r^ (cos {Q)f + ( - 2 r.4 -h 2 r^>l + 6 r^ + 6 r) cos {Q) - l - r ' ' A - r ^ + A-6r2. L e m m a 3.0,2. The polynomial R{x) has "^ AeR. 17
(3.18) (3.19)
" one real root smaller than one for ^
Proof: We begin by solving the polynomial using the quadratic formula. This yields the following roots:
3r' + .AT- + 3 - A + ^/br^ + 2 Ar^ - 6r^ + A^r^ - 2 A^r^ 4 - 5 - 2 A + A^ 4r and 3 r^ + Ar- + 3 - .4 - ^/b?^+ 2 Ar^ - 6 r^ + A^r^ - 2 A^r^ + b-2A Ar + A'^
,^
^Q)
.
Since r, A, and 9 are all real, we have that the roots of the polynomial occur in conjugate pairs. Hence, if the term under the radical is negative, both roots are complex and we are done. So we can assume the radicand is non-negative. If the radical is real, the lemma will hold if:
3^-2 ^ Ar- + 3-A
+ ^/5r^ + 2Ar^-Qr^ Ar
+ Ah^ - 2 A r ^ + 5 - 2 A + A ^ ^ (3.22)
We multiply through by Ar. Next we subtract Ar from both sides to get:
3^2 _^ ^^2 _ 4^ + 3 _ ^ + V5r4 + 2 Ar4 -Qr'' + A V - 2 A^r^ + 5 - 2 A -h A > 0 (3.23) Since the radical is non-negative, if Zr'+ Ar^ - Ar-^2> - A> 0, we will be done. Observe that if we take the left hand side as a linear function in A we get (r^ -\)A +
3r2 - 4r + 3 which is decreasing in A Thus, for a given r G (0, 1) the expression will have its minimal value on (-oo, 1] at A = 1. Substituting gives: Ar' - Ar+ 2, which is positive for all 0 < r < 1. Hence for A < 1 we have Sr^ + Ar^ - 4r + 3 - A > 0. Now we consider the case where A > 1. If 3r^ + Ar^ - 4r -^ 3 - A > 0, then we are done anyway. So assume the contrary '^v...acting Zr'' -h Ar^ - 4r + 3 - A from both sides of (3.23) gives 18
\/5r4 + 2 . - i H - G r M r T V ^ r T : 4 2 , 2 + 5 - 2 A - f A 2 > - ( 3 r 2 - h A r 2 - 4 r - F 3 - A ) (3.24) As the term inside the parenthesis is by hypothesis negative, the right hand side of 3.24 is positive. This give...