This has already been discussed at length in this forum, but since you asked, here goes.
The #1 best defense against bears is a good can of bear spray. I don't really care what anybody on here says; it's been studied and researched thoroughly here in Alaska and it has proven itself time and time again. It has proven to be 97%-100% effective (depends on which study you read) against bears, while handguns remain at or below the 60% mark.
That being said; I carry a 9oz can of bear spray and my Glock 20SF (10mm) loaded with 200gr WFNGC Beartooth rounds from Double Tap + two spare mags. Yes, I do really live in Kodiak. Yes I have had run-ins with a couple bears. And no, I haven't had to shoot one yet. Had to replace a few cans of bear spray that I've used (along with a few sets of underwear
) but I did not have to shoot. I draw them both at the same time just in case the spray does not affect the bear. I'm still alive with all my body parts still attached, so there must be something to this bear spray stuff...
Honestly, nothing short of a .375 H&H and higher will likely stop/kill a bear in it's tracks, but I ain't gonna tote a 10 pound bolt gun on my shoulder when I'm hiking, fishing, deer hunting, rabbit hunting, duck hunting, etc... I carry that 10mm because I am deadly accurate with it and it's so light that I won't have an excuse to leave it home..
Will the 10mm penetrate a Kodiak Brown Bear's skull and kill it? No, but neither will a .44 Mag and I don't care what type of bullet you're using; it just won't do it. I've seen and handled a Kodiak Brownie skull and it ain't no joke. When measured on a horizontal plane (the plane on which the bullet will be traveling) the skull measures roughly 4" or so- on a mature adult. So with a headshot, you're shooting something twice as hard as concrete, 4" thick and on about a 45 degree angle; all in an effort to hit the softball-sized brain? Not gonna happen with a handgun. Sorry to bust your bubble .44 Mag lovers, it just can't do it.
The best option (when forced to use your handgun
after unsuccessfully using bear spray) is to shoot and disable the front shoulders. If you shatter the front shoulders, it's mobility will be only heavily hampered/slowed at worst, and completely disabled at best. Thus giving you the time to reload if necessary, get to the side of the bear if possible and place an aimed kill shot to the side of the bear's head or spine. The sides are less protected allowing for one or two shots to the brain, thus ending the threat. (I discuss the 'kill shot' only because under Alaska state law, if you wound or otherwise seriously injure an animal, you must then track it down and kill it. The last thing we want is a ****ed off, wounded bear running around.)
Just about the only chance of a frontal kill shot to the head of a big brownie is to shoot through the nasal passage to the brain. Not a likely scenario to say the least.
You're best bet is bear spray. Carry a gun that you are inherently familiar with and can shoot accurately in the .357 Mag and up range for backup only.
Hope this helps.