Is there an equivalent of "Don't stroke his ego" in Japanese? I assume that such a phrase would be relevant in Japanese, but I've never heard it.
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There should be a dozen or more "equivalents". As a general tip, you will always receive a better answer if you let us know who is speaking to who -- ages, genders, relationship, etc. The more information you give, the better and more natural the phrases will be.
Off the top of my head, I could give you: