within the sphere of influence
Used in political, military, or economic contexts to describe an area or group under someone's control or influence. Often followed by particles like に or の.
That country is within the sphere of influence of a major power.
In areas within its sphere of influence, the company's influence is strong.
勢力圏 means 'sphere of influence' itself, while 勢力圏内 specifically means 'within that sphere'.
影響下 also means 'under the influence of', but is more general and can refer to any kind of influence, not necessarily a geopolitical or organizational sphere.
Compound of 勢力圏 (sphere of influence) and 内 (within). The exact historical derivation is uncertain, but the word follows a common pattern of adding 内 to a noun to mean 'within that area or scope'.