Friends any hint as to why is this set of equations a test of linearity of a derivative security?

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From Taleb - Dynamic Hedging pg. 11

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**,,** Derivatives are not always linear, convex, or concave across all moves (See Figures 1.2A-D, I did put a picture excerpt from the book down
). A test of local linearity of a derivative security (that is a function of the underlying asset) between asset prices $S\_1$ and $S\_2$ with $0<λ<1$, will satisfy the following equality: $$V(λS\_1 + (1-λ)S\_2) = λV(S\_1) + (1 - λ)V(S\_2)$$

It will be convex between $S\_1$ and $S\_2$ if: $$V(λS\_1 + (1-λ)S\_2) ≤ λV(S\_1) + (1 - λ)V(S\_2)$$
It will be concave if: $$V(λS\_1 + (1-λ)S\_2) ≥ λV(S\_1) + (1 - λ)V(S\_2)$$

**,,**

Any hint which mathematical theorem is behind this equation. $V$ i assume means Value of the derivative security. Is there somewhere to read to hone some insight about this for the unenlightened scholar?

[![enter image description here](https://i.stack.imgur.com/wbPj8.jpg)](https://i.stack.imgur.com/wbPj8.jpg)