<a name="docA001">
2010-03-10-10-00 start
This file Hölder's Inequality Curve
or        Holder's Inequality Curve
http://freeman2.com/holder02.htm 
is written during Liu,Hsinhan study
the book
The Cauchy-Schwarz Master Class
J. Michael Steele ★★★★★
<a name="docA002">
ISBN 978-0-521-54677-5
chapter 9 Holder's Inequality.
Start code from 2010-03-08-12-53
done  code  at  2010-03-10-00-20

<a name="docA003">　contralCenter　click button
Page control panel is next
[[
<a name="HolderBgn"> 2010-03-08-13-30 NO complex 
Graph area size, W:[300]  H:[300]  □complex 
x min:[0], x max:[20]; y min:[0], y max:[10]; 
x/y min/max : ▣auto scale, □user scale Click button 
10^[1.2345]; ▣+/0 , □+/0/-; [5]digits; □eq.p   Help1  
fill [20] numbers; □integer;   b=a^[non-proportion]  
Box 1, ak sequence     Box 2, bk sequence   
Box 3, output          Box 4, Dot     
Box 5,   
<a name="docA004">
<a name="Holder1401"> Hölder Inequality, Holder Inequality
===Hölder Inequality equation here===
require: 1/p + 1/q=1　[default][420][380] calling ID 1401 
p/s bgn[1.01], p/s end[20]  steps[80]  [ds]
[randomA] [d5] [u] [v] [randomB] Why 'A', 'B' ? 
Click to RUN [HolderA] [HolderB] [Wipe] control center 
]]

<a name="docA005">
Explain as following
[[
<a name="HolderBgn"> 2010-03-08-13-30 NO complex 
Graph area size, W:[300]  H:[300]  □complex 
]]
User can control graph board size by
specify W:[300]  H:[300] numbers.
User's parameter will become effective
only if checked "□user scale" checkbox.

<a name="docA006">
If checked "□complex" then click [randomA]
output complex random number to Box 1 & 2.
This program Holder Inequality Curve 
do not work with complex number. If you
input complex, program read real part 
only.

<a name="docA007">
[[
x min:[0], x max:[20]; y min:[0], y max:[10]; 
x/y min/max : ▣auto scale, □user scale Click button 
]]
User can control x-axis/y-axis range by
specify 
x min:[0], x max:[20]; y min:[0], y max:[10]; 
numbers.
<a name="docA008">　contralCenter　click button
If "▣auto scale" checked, Board size and
x/y min/max are determined by program.
If "□user scale" checked, Board size and
x/y min/max are determined by user.
"Click button" bring you to Click button
quickly.

<a name="docA009">
[[
10^[1.2345]; ▣+/0 , □+/0/-; [5]digits; □eq.p   Help1  
[rA] [rB] fill [20] numbers; □integer;   b=a^[non-proportion]  
]]
Above two lines control random number.
Random number function is provided for
test this program.
<a name="docA010">
10^[1.2345]; generate random number
between 0 and 10^1.2345=17.16
User can change [1.2345] value.
▣+/0 ,   if checked, random number is
         non-negative.
□+/0/-;  if checked, random number is
         positive/zero/negative.
<a name="docA011">
[5]digits; random number has five
         digits.
□eq.p    if checked, probability number
         has equal value.
Help1    goto help quickly.
[rA] [rB] are randomA and randomB.
fill [20] numbers; generate this many
         random numbers.
<a name="docA012">
□integer;  if checked, random number
         is integer.
b=a^[non-proportion]  if fill a number
         then bk seq. is proportional
         to ak sequence. Result is
         Holder's Inequality become
         equality. more

<a name="docA013">
[[　contralCenter　click button
Box 1, ak sequence     Box 2, bk sequence   
Box 3, output          Box 4, Dot 
Box 5,   
]]
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Box 1 let user fill in ak sequence
Box 2 let user fill in bk sequence
Box 3, output  has [p,f(p)] pair data
Box 4, Dot     has [p,f(p)] pair data
Box 5,   let user specify p value.
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Box 3 [p,f(p)] pair not controllable.
Box 4 [p,f(p)] pair is  controllable.
User input desired p value to Box 5.
find f(p) at Box 4.
Box 5 narrow to one line box. make
room for wider Holder Inequality 
equation.

<a name="docA016">
[[
<a name="Holder1401"> Hölder Inequality, Holder Inequality
===Hölder Inequality equation here===
require: 1/p + 1/q=1　[default][420][380] calling ID 1401 
]]
Holder Inequality equation printed
here for quick reference.
<a name="docA017">
[default][420][380] let user control
equation field width. If user use
page text font size different from
LiuHH programming time font size,
User can change [420][380] values.
[420] is outer frame width.
[380] is inner frame width.

<a name="docA018">
[[　contralCenter　click button
p/s bgn[1.01], p/s end[20]  steps[80]  [ds]
]]
[HolderA] draw 1/p + 1/q =1 curve
[HolderB] draw  s  +  t  =1 curve
[HolderA] use p as variable, 1＜p＜∞
[HolderB] use s as variable, 0＜s＜1
<a name="docA019">
User can control [HolderA] p value 
at p/s bgn[1.01], p/s end[20] boxes
User can NOT control [HolderB] s 
value. Default s in [0.005,0.995]
see code at
[[
<a name="docA020">
sDelta=0.005; //9903092032
if(b0==0)s=s+sDelta; //9903092033
else //avoid singular points
if(b0==step5)s=s-sDelta; //9903092034
]]

<a name="docA021">
steps[80] decide draw how many short
straight lines to create a smooth
curve. Default 80.
steps must be greater than 9. If you
input 10 for steps, you will see a
polygon type 'curve'.

[ds] button delete 
p/s bgn[], p/s end[]  steps[] 

<a name="docA022">
[[
[randomA] [d5] [u] [v] [randomB] Why 'A', 'B' ? 
]]
If random number simulate observed 
data, no need sum to one.
If random number simulate probability 
do need sum to one.
<a name="docA023">　contralCenter　click button
[randomA] create a-sequence and
          b-sequence, both not sum 
          to one
[randomB] create one a-sequence and
          one probability sequence.
          Box 2 sequence sum to one.
<a name="docA024">
This program Holder Inequality curve
need two array observed data, and do
not need probability. This program
need only [randomA] button.
[randomB] is provided let user
generate probability sequence
here and use it at other page.

<a name="docA025">
[d5] delete Box 5 test p values.
[u] generate test p values 1＜p＜inf.
[v] generate test s values 0＜s＜1
All test points will show up at 
Box 4 and show up on drawing as 
a red dot.

<a name="docA026">
[[
Click to RUN [HolderA] [HolderB] [Wipe] control center 
]]
[HolderA] draw 1/p + 1/q =1 curve
[HolderB] draw  s  +  t  =1 curve
[HolderA] use p as variable, 1＜p＜∞
[HolderB] use s as variable, 0＜s＜1
[Wipe] erase drawing
"control center" goto control center 
quickly
2010-03-10-11-30 here

<a name="docA027">
2010-03-10-11-48
Why 'A', 'B'
"randomA"  no probability
"randomB" has probability
[HolderA] draw 1/p + 1/q =1 curve
[HolderB] draw  s  +  t  =1 curve
2010-03-10-11-52 stop

<a name="docA028">　contralCenter　click button
2010-03-10-17-15 start
This program local
http://freeman2.com/holder02.htm
draw Holder's Inequality curve.

<a name="docA029">
Holder's Inequality and 
Power Mean Inequality and
Cauchy's Inequality all
look alike. Inequality new-hand 
reader may not be clear the
differences between above three
inequalities. 
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The following is a comparison.
First, please see Cauchy's Inequality

<a name="docA032">
Holder and Cauchy two inequalities
are the same. Holder is general.
If in Holder set p=2 and set q=2
satisfy 1/p + 1/q = 1 = 1/2 + 1/2
then Holder become Cauchy.
(sum to infinity or sum to n both
 are OK, not important)
<a name="docA033">　contralCenter　click button
Next is Power Mean Inequality.

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Power Mean Inequality and
Holder inequality are different.

Power Mean need two sequences, 
one is observed data, second is 
probability. Probability must
sum to one.
Holder Inequality need two 
sequences, both are observed 
data. both are NOT sum to one.

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Power Mean need one set 
observed data.
Holder Inequality need two 
sets observed data.

Before summation, observed data
has power operation.
Before summation, probability 
do not have power operation.

<a name="docA036">
Power Mean greater than side and
less than side, both have same
power operation (but different
number)
Holder Inequality less than side
do not have power operation. 
Only greater than side has power 
operation.

<a name="docA037">
Power Mean function value is 
monotone increase with power t.
Holder Inequality function value 
first decrease then increase when
power p increase.

<a name="docA038">　contralCenter　click button
If probability always sum to one,
Power Mean function curve do
not have singular point. Power 
Mean Inequality is a smooth curve.
Holder Inequality function curve 
has two singular points, p=0 or
p=infinity.

<a name="docA039">
Please get the following file
Power Mean Inequality Curve local
http://freeman2.com/powmean2.htm

Compare two programs holder02 and
powmean2. They will help you to
understand the difference of two
kind inequalities.

<a name="docA040">
Liu,Hsinhan try to write program
give correct output/drawing. But
no similar program for reference
and no one proofread/proofrun.
LiuHH can not promise output and
drawing are correct. If you find 
bug/error in output/drawing and
if you know programming, you are
welcome to modify/correct these
programs. 
<a name="docA041">
If you do not know 
programming, please come back to
freeman2.com/freeman2.htm#whatsnew
and check updates. If LiuHH find
error LiuHH will make correction 
and update file.

<a name="docA042">
This file holder02.htm no direct
way to verify output/drawing is
correct or not. But special case
do give reasonable output. This
special case is 
the case Holder Inequality become
equality.
<a name="docA043">　contralCenter　click button
Textbook page 137 and 138.
Please go to contralCenter
change 
from "b=a^non-proportion"
  to "b=a^ 2"
change 
from  step=80
  to  step=222
then click "HolderB" button.
<a name="docA044">
Drawing-below-document say
[[
Curve minimum occur at s=1/p=0.3333333333333333
p=3 and
minimum value is 1421.274084
]]
for given r=p/q, calculation
show that p=r+1 has equality 
(and minimum) Drawing output 
do confirm this point.

<a name="docA045">
"HolderB" button use s=1/p as
x-axis variable, from s=0 to 
s=1 for whole range.

"HolderA" button use p as
x-axis variable, from p=1 to 
p=infinity for whole range.

<a name="docA046">
"HolderB" step on min point
easier than "HolderA". So
please try "HolderB".

Above is only way to verify
holder02.htm
LiuHH hope output/drawing are
correct.

Thank you for visiting Freeman's 
web page.　 
2010-03-10-18-13

<a name="docA047">
[[　contralCenter　click button
If you want to keep drawing.
Use keyboard "copy screen"
function. Paste full screen
image to paint.exe and save
to file.
2010-02-22-18-53 stop
]]

<a name="docA048">
2010-03-15-13-06 start
"Update 2010-03-15" change as 
following
(1) add link to "10^" and "b=a^"
(2) add link to "calculation" in
    [[docA044
    for given r=p/q, calculation
    show that p=r+1 has equality 
    (and minimum) Drawing output 
    do confirm this point.
    ]]
<a name="docA049">
(3) change "HolderB" button.
    before change, "HolderB" domain
    lock at s=[0.005,0.995]
    after  change, "HolderB" domain
    can be assigned by user
    if  p/s bgn  , p/s end 
    value is within [0,1] and 
    if s bgn ＜ s end
<a name="docA050">
Older version lock domain at 
s=[0.005,0.995] that is because
LiuHH think s range from 0 to 1
much better than p range from 1 
to infinity. 0 to 1 no need to 
change. But, 
<a name="docA051">
later LiuHH want to
narrow down the domain to minimum
value point, there is no way to 
do so in "HolderB" button. Because
LiuHH locked range at [0.005,0.995]
From code time stamp 9903151156 
to 9903151212 change "HolderB" 
behavior. 
<a name="docA052">
Allow user to specify
s domain. If p/s bgn  , p/s end
value is in [0,1] program will
accept it. For [0,1] input, user
will see singular phenomenon at
two ends.
2010-03-15-13-33 stop