One function is greater than other

I would like to determine the range of values for \( z \) where the following inequality holds true:

this is my trying

syms z real
assume(z > exp(1))
% Define the function
f =  z - 8.02 * log(z) - (3.359 / 21.233) * log(z) * z;

sol = solve(f > 0, z, 'ReturnConditions', true);

Warning: Unable to find explicit solution. For options, see help.

vpa(sol.conditions)

 
ans =
 
Empty sym: 0-by-1
 

Answer 1

[Matt J]

Let us reorganize the inequality as,

Since log(z)>0 in the given domain of z, this implies which in turn implies that log(z)<1/0.1582, or in turn that z<556.19. Therefore, if the inequality is ever going to be satisfied, it must be satisfied somewhere in the interval [e,556.2]. We can see from plotting that this never occurs,

syms z real
fplot(z, [exp(1),556]); hold on
fplot(log(z).*(0.1582*z+8.02), [exp(1),556.2]); hold on; legend(Location='NW')

but we can also check it with fminbnd,

[zmin,fmin]=fminbnd(@(z) log(z).*(0.1582*z+8.02)-z  ,exp(1),556.2)

zmin = 143.8330

fmin = 9.0742

Answer 2

[Walter Roberson]

Q = @(v) sym(v);

syms z real
assume(z > exp(1))
% Define the function
f =  z - Q(802)/Q(10)^2 * log(z) == (Q(3359)/Q(10)^3 / (Q(21233)/Q(10)^3)) * log(z) * z;
F = lhs(f) - rhs(f)

F = 

limit(F, z, inf)

ans = 

fplot(F, [exp(1), 1000])

fplot(lhs(f), [exp(1), 200])
hold on
fplot(rhs(f), [exp(1), 200])
legend({'lhs', 'rhs'})

There is no solution. Everywhere in the range, the left hand side is less than the right hand side.

[Fatima Majeed]

But you just do it for z in [exp(1),200]

[Walter Roberson]

You have stated

assume(z > exp(1))

so we are justified in starting from exp(1) in the fplot()

[Walter Roberson]

I do not prove that there are no crossings... but I give evidence.

Q = @(v) sym(v);

syms z real
assume(z > exp(1))
% Define the function
f =  z - Q(802)/Q(10)^2 * log(z) == (Q(3359)/Q(10)^3 / (Q(21233)/Q(10)^3)) * log(z) * z;
F = lhs(f) - rhs(f)

F = 

limit(F, z, inf)

ans = 

The limit of the difference is -infinity, so there must be an even number of zero crossings (no zero crossings counts as an even number of them.)

fplot(lhs(f), [exp(1), 1000000])
hold on
fplot(rhs(f), [exp(1), 1000000])
legend({'lhs', 'rhs'})

The two are getting further apart up to at least 1 million.

Additional logic would be needed to establish that there are definitely no crossings.

Answer 3

[Sam Chak]

Hi @Fatima Majeed

The search below covers this range .

z   = linspace(0, exp(1), 3001);
fun = @(z) z - 8.02*log(z) - (3.359/21.233)*log(z).*z;
plot(z, fun(z)), grid on

sol = fzero(fun, 1)

sol = 1.1506

f1  = z - 8.02*log(z);             % LHS of inequality
f2  = (3.359/21.233)*log(z).*z;    % RHS of inequality

plot(z, [f1; f2]), grid on, ylim([-0.2 2]), xlabel('z')
txt = sprintf('%.4f', sol);
xline(sol, '--', txt)
legend({'$z - 8.02 \log(z)$', '$\frac{3.359}{21.233} \log(z) z$', ''}, 'interpreter', 'latex', 'fontsize', 14);

Thus, the inequality holds at .