Verification of Inequalities
(i) Four practical mechanisms
The role of CAS in analysis
(ii) Applications
Kent Pearce
Texas Tech University
Presentation: January 2008
Question
 Given a function f on an interval (a, b), what
does it take to show that f is non-negative on
(a, b)?
 Proof by Picture

Maple, Mathematica, Matlab, Mathcad,
Excel, Graphing Calculators
(P)Lots of Dots
(P)Lots of Dots
(P)Lots of Dots
(P)Lots of Dots
y  f ( x) 
1
2x 1
(P)Lots of Dots
y  f ( x) 
1
2x 1
Blackbox Approximations
 Polynomial
Blackbox Approximations
 Transcendental / Special Functions
Practical Methods
 A. Sturm Sequence Arguments
 B. Linearity / Monotonicity Arguments
 C. Special Function Estimates
 D. Grid Estimates
Applications
 "On a Coefficient Conjecture of Brannan," Complex
Variables. Theory and Application. An International
Journal 33 (1997) 51_61, with Roger W. Barnard and
William Wheeler.
 "A Sharp Bound on the Schwarzian Derivatives of
Hyperbolically Convex Functions," Proceeding of the
London Mathematical Society 93 (2006), 395_417, with
Roger W. Barnard, Leah Cole and G. Brock Williams.
 "The Verification of an Inequality," Proceedings of the
International Conference on Geometric Function Theory,
Special Functions and Applications (ICGFT) (accepted)
with Roger W. Barnard.
 "Iceberg-Type Problems in Two Dimensions," with
Roger.W. Barnard and Alex.Yu. Solynin
Practical Methods
 A. Sturm Sequence Arguments
 B. Linearity / Monotonicity Arguments
 C. Special Function Estimates
 D. Grid Estimates
Iceberg-Type Problems
Iceberg-Type Problems
 Dual Problem for Class 0
Let D= {z  : 0  | z |  1} and let
1

  { f ( z )   a0  a1 z 
: f is analytic,
z
univalent on D}. For f   let E f  \ f (D)
and 0  { f   | 0  E f }. For 0 < h < 4, let
H h  {z | Re( z )  h}. Find
A(h)  max area( E f  H h )
f 0
Iceberg-Type Problems
 Extremal Configuration




Symmetrization
Polarization
Variational Methods
Boundary Conditions
Iceberg-Type Problems
Iceberg-Type Problems
 We obtained explicit formulas for A = A(r)
and h = h(r). However, the orginial problem was
formulated to find A as a function of h, i.e. to find
A = A(h).
 To find an explicit formulation giving A = A(h),
we needed to verify that h = h(r) was monotone.
Sturm Sequence Arguments
 General theorem for counting the number of
distinct roots of a polynomial f on an interval
(a, b)
 N. Jacobson, Basic Algebra. Vol. I., pp. 311315,W. H. Freeman and Co., New York, 1974.
 H. Weber, Lehrbuch der Algebra, Vol. I., pp. 301313, Friedrich Vieweg und Sohn, Braunschweig,
1898
Sturm Sequence Arguments
 Sturm’s Theorem. Let f be a non-constant
polynomial with rational coefficients and let a < b
be rational numbers. Let S f  { f 0 , f1 , , f s }
be the standard sequence for f . Suppose that
f (a)  0, f (b)  0. Then, the number of distinct
roots of f on (a, b) is Va  Vb where Vc denotes
the number of sign changes of
S f (c)  { f 0 (c), f1 (c),
, f s (c)}
Sturm Sequence Arguments
 Sturm’s Theorem (Generalization). Let f be a
non-constant polynomial with rational coefficients
and let a < b be rational numbers. Let
S f  { f 0 , f1 , , f s } be the standard sequence for
f . Suppose that f (a)  0, f (b)  0. Then, the
number of distinct roots of f on (a, b] is Va  Vb
where Vc denotes the number of sign changes of
S f (c)  { f 0 (c), f1 (c),
, f s (c)}
Sturm Sequence Arguments
 For a given f, the standard sequence S f is
constructed as:
f0  f
f1  f 
f 2 : f 0  f1q1  f 2
f 3 : f1  f 2 q2  f 3
Sturm Sequence Arguments
 Polynomial
Sturm Sequence Arguments
 Polynomial
Linearity / Monotonicity
 Consider
f ( x, Z )  c0 ( x)  c1 ( x)Z
where   Z  
Let
f ( x)  f ( x, Z ) Z   c0 ( x)  c1 ( x) ,
f  ( x)  f ( x, Z ) Z    c0 ( x)  c1 ( x) 
Then,
min { f ( x), f  ( x)}  f ( x, Z )  max{ f ( x), f  ( x)}
x( a ,b )
x( a ,b )
Iceberg-Type Problems
 We obtained explicit formulas for A = A(r)
and h = h(r). However, the orginial problem was
formulated to find A as a function of h, i.e. to find
A = A(h).
 To find an explicit formulation giving A = A(h),
we needed to verify that h = h(r) was monotone.
Iceberg-Type Problems
 From the construction we explicitly found
where
Iceberg-Type Problems
Iceberg-Type Problems
where
Iceberg-Type Problems
 It remained to show
g  g (r )  (c0  c1P)  (d0  d1P)Q
was non-negative. In a separate lemma, we
showed 0 < Q < 1. Hence, using the linearity of
Q in g, we needed to show
g 0  (c0  c1P)  (d 0  d1P)  0
g1  (c0  c1 P)  (d 0  d1P) 1
were non-negative
Iceberg-Type Problems
 In a second lemma, we showed s < P < t where
Let
g0,s  g0 Ps , g0,t  g0 Pt , g1, s  g1 Ps , g1,t  g1 Pt .
Each g0, s , g0,t , g1, s , g1,t is a polynomial with
rational coefficients for which a Sturm sequence
argument show that it is non-negative.
Practical Methods
 A. Sturm Sequence Arguments
 B. Linearity / Monotonicity Arguments
 C. Special Function Estimates
 D. Grid Estimates
Notation & Definitions

D  {z :| z |  1}
Notation & Definitions

D  {z :| z |  1}
hyperbolic metric
2 | dz |
 ( z ) | dz |
2
1 | z |
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
 Hyberbolically Convex Function
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
 Hyberbolically Convex Function
 Hyberbolic Polygon
o Proper Sides
Examples
k ( z ) 
2 z
(1  z )  (1  z ) 2  4 2 z
k
Examples

f ( z )  tan  

where  
z
 (1  2
2
1
4  2
cos 2   )
0

2 , 0  
2
K (cos )
f

d 

Schwarz Norm || S f ||D
For f  A(D) let

Sf  

2

f   1  f  
  

f   2 f  
and
|| S f ||D  sup{D2 ( z) | S f ( z) |: z D}
where
1
D ( z ) 
1 | z |2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f (D) convex  || S f ||D  2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f (D) convex  || S f ||D  2
 Spherical Convexity

Mejía, Pommerenke (2000):
f (D) convex  || S f ||D  2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f (D) convex  || S f ||D  2
 Spherical Convexity

Mejía, Pommerenke (2000):
f (D) convex  || S f ||D  2
 Hyperbolic Convexity

Mejía, Pommerenke Conjecture (2000):
f (D) convex  || S f ||D  2.3836
Verification of M/P Conjecture
 "A Sharp Bound on the Schwarzian Derivatives
of Hyperbolically Convex Functions,"
Proceeding of the London Mathematical Society
93 (2006), 395_417, with Roger W. Barnard,
Leah Cole and G. Brock Williams.
 "The Verification of an Inequality," Proceedings
of the International Conference on Geometric
Function Theory, Special Functions and
Applications (ICGFT) (accepted) with Roger W.
Barnard.
Verification of M/P Conjecture
 Invariance under disk automorphisms
 Reduction to hyperbolic polygonal maps
 Reduction to | S f (0) |
 Julia Variation
 Reduction to hyperbolic polygonal maps with at
most two proper sides
 Reduction to
(1  x2 )2 | S f ( x) |, 0  x  1, 0    1
 Reduction to max S f (0)
0  1
Graph of S f
 ( )
*
(0)  2(c   )
2
Two-sided Polygonal Map

f ( z )  tan  

where  
z
 (1  2
2
1
4  2
cos 2   )
0

2 , 0  
2
K (cos )
f

d 

Special Function Estimates
 Parameter  
 /2
K ( y)
where y  cos 
Special Function Estimates
 Upper bound
Special Function Estimates
 Upper bound
 Partial Sums
Special Function Estimates
(1 | z |2 )2 | S f ( z) |
θ = 0.3π /2
(1 | z |2 )2 | S f ( z) |
θ = 0.5π /2
Verification of M/P Conjecture
 Invariance under disk automorphisms
 Reduction to hyperbolic polygonal maps
 Reduction to | S f (0) |
 Julia Variation
 Reduction to hyperbolic polygonal maps with at
most two proper sides
 Reduction to
(1  x2 )2 | S f ( x) |, 0  x  1, 0    1
 Reduction to max S f (0)
0  1
Verification
where c  cos 2 ,  
 /2
, 1  x  1
K (cos  )
Graph of c  
2
0
Verification
where c  cos 2 ,  
 /2
, 1  x  1
K (cos  )
Verification c3  0
Straightforward to show that q p  0
In qm make a change of variable c  2 y  1
2
Verification c3  0
Obtain a lower bound for qm by estimating 
via an upper bound
*
q
Sturm sequence argument shows m  qm   p8
is non-negative
Grid Estimates
Grid Estimates
 Given


A) grid step size h
B) global bound M for maximum of | f ( x) |
 Theorem Let f be defined on [a, b]. Let
M  max | f ( x) | . Let   0 and suppose that N is
x[ a ,b ]
chosen so that h  (b  a) / N   . Let L be the
f ( x)
lattice L  {a  jh : 0  j  N } . Let m  min
xL
If
 
m  M  ,
2
then f is non-negative on [a, b].
Grid Estimates
 Maximum descent argument
Grid Estimates
 Two-Dimensional Version
Grid Estimates
 Maximum descent argument
Verification
where c  cos 2 ,  
 /2
, 1  x  1
K (cos  )
Verification c2
 The problem was that the coefficient c2 ( , x) was
not globally positive, specifically, it was not
positive for 1  x   4 5 , 0     2 .
 We showed that p4 (t )  0 by showing that
q(t )  0 where
q(t )  c3 ( , x)t 2  c2 ( , x)t  c1 ( , t )
0 < t < 1/4.
Verification c2
 For the case,  4     2 expand q(t) in powers of 
q(t )  c4 (c, x, t ) 4  c2 (c, x, t ) 2  c0 (c, x, t ).
Noting that c4 (c, x, t ) and c2 (c, x, t ) are negative,
we replaced  by an upper bound ( of 1)
to obtain a lower bound
q(t )  q0 (t )  q(t )  1  e2 ( y, x)t 2  e1 ( y, x)t  e0 ( y, x)
where y  cos
Verification c2
 Finally, we introduced a change of variable
to obtain
q0* (t )  q0 (t ) x 1 2 w /10  e2 ( y, w)t 2  e1 ( y, w)t  e0 ( y, w)
where the coefficients are polynomials (with
rational coefficients) in w, y and
0  w  1, 0  y  2
2
, 0t  1
4
Verification c2
 Used Lemma 3.3 to show that the endpoints
q0* (0)  e0 ( y, w) and q0* ( 1 4) are non-negative. We
partition the parameter space into subregions:
Verification c2
 Application of Lemma 3.3 to q0* ( 1 )
4
 After another change of variable, we needed to
show that r  0 where
for 0 < w < 1, 0 < m < 1
Verification c2
Verification c2
[0,1]  [0,1]  [0,1/ 2]  [0,1/ 2]  [0,1/ 2]  [1/ 2,1]
[1/ 2,1]  [0,1/ 2]  [1/ 2,1]  [1/ 2,1]
Quarter Square [0,1/2]x[0,1/2]
M  max{M w , M m }  35
M w  21, M m  35
Grid 50 x 50
m  min r ( w j , mk )  0.400
( w j , mk )L
M   0.350
Verification c2
 Application of Lemma 3.3 to non-negativity of
q0* (t ) on the subregion D:
 We showed that the discriminant of a related
quadratic function was negative on D. That
computation amounted to showing that a
polynomial hl (m, v), of degree 16 in m and
degree 40 in w, was non-negative
Conclusions
 There are “proof by picture” hazards


CAS numerical computations are rational number
calculations
CAS “special function” numerical calculations are
inherently finite approximations
 There is a role for CAS in analysis
 There are various useful, practical strategies for
rigorously establishing analytic inequalities