E(x)=(sinx/sqrt(x))+(cosx/sqrt(x+1)) . Prove that modulus of E(x) 0.
https://www.google.com/search?q=plot+sin%28x%29%2Fsqrt%28x%29%2Bcos%28x%29%2Fsqrt%28x%2B1%29%2C+sqrt%28%28x%2B1%29%2Fx%29
please, see it, both plots for x>0
As it is seen in the graph provided by Steftcho P. Dokov, the inequality is actually strict; i.e., |sinx/sqrt(x)+cosx/sqrt(x+1)|0. An algebraic proof of that is the following:
Let the plane vectors a=(sinx,cosx) and b=(1/sqrt(x),1/sqrt(x+1)).
If |a| and |b| are the norms of the two vectors, we have |a|=1 and |b|=sqrt(1/x+1/(x+1))=sqrt((2x+1)/(x(x+1))). Also, |ab|=|sinx/sqrt(x)+cosx/sqrt(x+1)|. Further, we have |ab|=|a||b||cos(a,b)|
Spiros Konstantogiannis , Steftcho P. Dokov Thank you very much for your interest in this problem, for your solutions too!
Spiros Konstantogiannis Congratulations dear Spiros, your solution is identical to the one given by me, i.e. using Schwartz inequality in R2. I'm glad you learned very solidly everything you saw in the past on RG!
Indeed, for x>0 we have
[(sinx/sqrt(x))+(cosx/sqrt(x+1))]2
As you can see above, Schwartz inequality in Hilbert space, applied contextually in a space Rn for some n>1 or in L2, can be a strong tool in solving nice but difficult elementary inequalities.
Dear Dinu and Dear Steftcho, the following statement is a small generalization to the given inequality.
For every point (x,y) on the unit circle x2+y2=1 and for every point (z,t) on the hyperbola z2-t2=a>=1 other than the two vertices (+-sqrt(a),0), it holds that |x/z+y/t|0.
Proof
Considering the plane vectors (x,y) and (1/z,1/t) and applying again the Cauchy-Schwarz inequality, we get |x/z+y/t|=0, we have that (3) holds; thus, (2) holds, as well. Finally, combining (1) and (2), we obtain |x/z+y/t|
This is a great move Spiros. Thanks. Perhaps, now, a generalization in 3-d could be done along such lines:
For every point (x,y,z) on the unit sphere x^2+y^2+z^2=1 and for every point (w,u,v) on the hyperboloid w^2-u^2-v^2=a>=1 other than the two vertices (+-sqrt(a),0,0), it hold |x/w+y/u+z/v| < w^2 / |(uv)|.
Spiros Konstantogiannis , Steftcho P. Dokov Many thanks for your valuable contributions! As I above said, Cauchy-Schwarz inequality helps us to solve or to generate many interesting inequalities!
Spiros Konstantogiannis Some interesting facts about this famous inequality( from where we have different writings of name Schwarz-Schwartz):
It seems Cauchy knew only the numerical inequality, i. e.
(SUM i=1,n a(i)b(i))2
I must say that I knew that Schwarz and Schwartz were surnames of different mathematicians; the first one being a German and the second being a contemporary French (with contributions to distributions), but I did not know that they both were involved in the said inequality. I thought only the German (Schwarz) was involved. In mathematical physics textbooks, the inequality is always reffered to as Cauchy-Schwarz inequality.
Regarding the authorship of some ideas in science, errors have often occurred and still occur. How many know that the theory of relativity and the notion of space-time were initiated by Hermann Minkowski, how many know that the initiator of the big bang theory was Georges Lemaitre? And I mean people from the math or physics area!