Can you activate a planeswalker's loyalty ability if it was put onto the battlefield under your control this turn?

Planeswalkers are not creatures, so the 'summoning sickness' rule (CR 302.6) that prevents creatures from attacking or using tap abilities does not apply to them. A planeswalker that just entered the battlefield under your control this turn can immediately have its loyalty abilities activated, provided you meet the normal timing restrictions.

Loyalty abilities are a special category of activated abilities. Under CR 606.3, a loyalty ability can only be activated during your main phase, when the stack is empty, when you have priority, and only once per turn per planeswalker. You also cannot activate a loyalty ability if doing so would cause the planeswalker's loyalty to drop below zero (CR 606.5).

Additionally, you may only activate one loyalty ability of a given planeswalker per turn (CR 606.4), regardless of how many abilities it has. But there is no restriction based on how recently the planeswalker entered the battlefield.

Example: You cast Liliana of the Veil on your turn. She enters the battlefield with 3 loyalty counters. You are now in your main phase with the stack empty — you can immediately activate her +1 ability to make each player discard a card, even though she just arrived this turn.