The study suggests that if these girls grow up believing tha

完整的句子是：The study suggests that 【if these girls grow up ( believing that boys are better at math than girls are),then these girls may not do as well as they would have if they were more confident.】
【】内为suggest的宾语从句.If these girls grow up...,为条件状语从句； believing that boys...为现在分词短语作伴随状语,修饰前面的these girls grow up.
宾语从句的主干句是then 后面的句子.

研究启示（暗示）假如这些女孩子在成长过程中相信男生在数学方面强于女生.....（由于主句没提及，很难确切表达suggest的意思）
believing that boys are better at math than girls are为现在分词词组作grow up的伴随状语，相当于grow up and believe that boys are better at math than...

grow up 是if 假设条件从句中的谓语动词，believing是伴随状语然后引导一个that从句,而这个that从句是believing的宾语从句

grow up 在从suggest引导的宾语从句中作谓语，believing 在此处为非谓语动词。
可翻译为：研究表明，是否这些女孩们在成长过程中都相信男生比女生数学成绩好。

希望能帮到你。