Expert: Paul Klarreich - 10/17/2007

Question
QUESTION: I believe the following inequality is true, BUT I haven't found a
reference nor I have been able to prove it...

I wonder if anyone could help me prove it or find a reference...Thanks

|sin ax| >= a |sin x| for all real x and all positive real a<=1.

[Difficulty]
I know that the following inequality is true. I do believe it may help prove the previous
one... Thanks

n |sin x|>= |sin nx| for all reals x and positive integers n

ANSWER: Questioner:   Guillermo
Category:  Advanced Math
Private:  No
 
Subject:  Trigonometric inequality
Question:  I believe the following inequality is true, BUT I haven't found a reference nor I have been able to prove it...

I wonder if anyone could help me prove it or find a reference...Thanks

|sin ax| >= a |sin x| for all real x and all positive real a <= 1.

[Difficulty]
I know that the following inequality is true. I do believe it may help prove the previous one... Thanks

n |sin x| >= |sin nx| for all reals x and positive integers n
..............................................
Hi, Guillermo,

There might be a good reason why you are having difficulty proving that  

|sin ax| >= a |sin x| for all real x and all positive real a <= 1.

I DON'T THINK IT IS TRUE.  I tried a quick and dirty graph of, for

example:

| sin (.217x) |  versus  .217 | sin x |

The first one is higher than the second MOST of the time, but not

always.  Only in the case  a = 1/n, where n is an integer, will it be true.  In that case, you shouldn't have any [Difficulty] using that known inequality.


---------- FOLLOW-UP ----------

QUESTION: Thanks for your answer. I am not fully convinced that

|sin ax| >= a |sin x| for all real x and all positive real a <= 1.

ONLY holds for a=1/n with n an integer...

Are we able to establish bounds for x or a in order to make the inequality hold???

Anyway, thanks a lot for your fast answer...

Answer
Questioner:   Guillermo
Category:  Advanced Math
Private:  No
 
Subject:  Trigonometric inequality
---------- FOLLOW-UP ----------

QUESTION: Thanks for your answer. I am not fully convinced that

|sin ax| >= a |sin x| for all real x and all positive real a <= 1.

ONLY holds for a=1/n with n an integer...

Are we able to establish bounds for x or a in order to make the inequality hold???

Anyway, thanks a lot for your fast answer...
..................................
Hi, Guillermo,

In my opinion, we are not to establish 'bounds' for a.  If you look at the graphs of the left and right sides, for some value of  a < 1, you see that each one looks like a 'bouncing ball', but they bounce at different rates and different heights.

|sin ax|  -- is SLOW, but higher.

a |sin x| -- is FAST, but lower.

If  a is exactly 1/n, for integer n, the zeroes (the bounce points?) of SLOW are a subset of the zeroes of FAST.  If, for example,  a = 1/3, then every third zero of FAST is a zero of SLOW.

But if a is not 1/n, then the zeroes are at different points.  Then the SLOW graph (the higher one) will always have some points -- its zeroes -- where it is lower than the FAST one.

.....................

Bounds for x?  That's different.  If you restrict x to this interval:

-pi/2 <= a <= pi/2

then I think your inequality will hold.  [But please don't hold your breath waiting for me to supply a proof.]