I do not know of research on this specific question that you are talking about ... probably because when you do research on people, it's hard to get that specific because the answer will usually be "it depends on the family situation and the people involved."

It might be a situation where you need to explore family counseling or professional help in getting everyone on the same page.

What we know about research into parenting is that "authoritarian" is more effective than "authoritative." The difference is that authoritative sets the rules, full stop. Authoritarian is not permissive, but is interested in hearing the child's perspective and incorporating them in developing the family standards. So what I would suggest is talking with your children about the research on social media/smartphone dangers. Explain your reasons for why you have concerns and feel you need standards. Talk about the dangers, the effects on mental health, etc. Also share data on how much usage is really out there ... teens think "everyone has a phone with no restrictions" but the reality is much different ... most parents have some restrictions, the questions are which restrictions.

Anyway, explain to them your concerns, listen to theirs, and work together on a plan and family standards. You're still the parent and can say "there are some things we can't negotiate on" for example, content filtering and limits on social media. But maybe you compromise, and then have stipulations ... the compromise is based on you following the rules (for example, no phones during family time/social times). We had to do this. Our first child just "gets it" and knows the dangers of social media and isn't that interested in it. She mostly just wants a phone that can call and text, which is easy to do without data plans. My second daughter is the fighter who pushed us on it. Like your child, she feels left out, and like everyone else has more freedom than her (although she later tells us her best friends have much tougher restrictions than we do and no wifi device at all! She hid that info from us originally. So realize they are only sharing info with you about the friends who have situations they want for themselves ... not the others).

Anyway, so we talked and heard her concerns. For her some things were more important about a phone than others. So we compromised and turned on some apps, as long as other apps could be off (you can have instagram and a camera, but not both at the same time ... just our way of slowing it down so she has to ask to switch her privilege from one to another). We also compromised by letting her have some apps/accounts, but limiting the amount of time she can spend on them each day. That negotiation continues, but the nice thing with parental controls in Apple and Android is you can adjust the amount of time for various apps.

It's been working better ... and she's a sophomore now so she's older. We had a hard rule about none of this stuff in middle school (but even then we compromised by helping her set up group me/hangouts for web-based texting on a desktop). So we're compromising more with her in high school to give her more freedom.

But as I mentioned above ... we regret some of those compromises and see the negative effects on her mental well being. So it's not the perfect situation I would like, but at least we have a good relationship with her. So sometimes you have to make those compromises. We plan on having a higher standard with our next child in the hope he'll be more like our first child and not push back on us so much. Because ideally we would have waited longer

Final thoughts are just to pray. a lot. Parenting teens is super tough.