$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $

I have noticed experimentally that the following question has a positive answer.

Is it true that for all $f, g$ even and convex functions $f$, $g$:

$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x) dx \times \int_0^1 g(\cos(1/x))dx? $$$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $$

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edited Mar 4, 2022 at 14:33

Dattier

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asked Mar 4, 2022 at 12:58

Dattier

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$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x) dx \times \int_0^1 g(\cos(1/x))dx? $

I have noticed experimentally that the following question has a positive answer.

Is it true that for all $f, g$ even and convex functions :

$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x) dx \times \int_0^1 g(\cos(1/x))dx? $$

fa.functional-analysis convexity

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$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $

$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x) dx \times \int_0^1 g(\cos(1/x))dx? $

$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $

$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x) dx \times \int_0^1 g(\cos(1/x))dx? $