Now just so everyone know, I've only watched the shows. I've seen every movie and every episode of Inuyasha. But I haven't read the Manga so do not be upset with me if some of my thoughts are different.
I believe that Sesshomaru, at any point in time could have killed Inuyasha. Yes, their father did leave Tenseiga to Sesshomaru and Tetsusaiga to Inuyasha, making it futile for them to fight. But look at everything that happened.
In the early episodes, Sesshomaru used a human arm from Naraku and showed Inuyasha the Wind Scar. He could've easily just used the Wind Scar on Inuyasha.
The episode where Sesshomaru uses the Dragon Arm, he tells Inuyasha about the Wind Scar, and fights Inuyasha until Inuyasha was able to finally see it and use it to defeat Sesshomaru (Tenseiga protected Sesshomaru from the blast and Sesshomaru meets Rin and his heart starts to grow.)
When Inuyasha became a full demon to kill Goshinki (Tokijin is created from his fangs.) Sesshomaru smelled his blood, went to the tree that created the Tetsaiga and Tenseiga's sheaths, and that tree explained things to Sesshomaru. Sesshomaru then finds Inuyasha killing a bunch of humans (while in demon form) and Sesshomaru says "Know your place, a half demon should act like a half demon" And when Miroku asked him if he gained feelings for his brother he replied "I will kill Inuyasha, but why will I kill him now when he doesn't know who he is." -In my opinion, Sesshomaru just wanted to stop Inuyasha's transformation because he didn't want his little brother to become a mindless killing demon, then he would HAVE to kill him.
Panther Demon tribe. When going through the past, Sesshomaru doesn't scoff at the idea of Inuyasha helping him the first time against the Panther Demon tribe. He said "Inuyasha doesn't want to help his older brother?" then they work with each other to defeat the Great Panther Demon. The fact that they started fighting each other (If you have a sibling, you often fought with one another) Sesshomaru was upset that Inuyasha had gotten so much stronger with the Tetsaiga, so he went to fight his younger brother. But eventually, they teamed up and beat the Demon.
Sesshomaru and Inuyasha fought Naraku together at the castle. While people think that Sesshomaru would've been engulfed by Naraku, you are wrong, Sesshomaru is always calm, and calculated, he would've escaped eventually. But yes Inuyasha did come and "save" Sesshomaru and they fought together and each other and worked together to eventually defeat Naraku.
I know this is long so bare with me, but there are a lot of episodes and I haven't gotten to the movie yet.
When Rin was attacked by Naraku big bees, I'm not spelling that darn name, what is it? Simioso? Or whatever, his bugs. Sesshomaru chased two down that had stung Jaken, so they died right at Inuyasha's feet and Sesshomaru was pissed because he though Inuyasha did it. Sesshomaru defeated Inuyasha rather easily, and could have killed him, but didn't, learned about the Ox Tiger and went on his way.
Sesshomaru punches Inuyasha when they are at their father's grave because he feels like Inuyasha was disrespecting their father's tomb. Then goes and fights Naraku.
In the Final Act, Sesshomaru builds up a technique called Meido Zangetsuha, then he makes Inuyasha fight him to earn it to get it. He then saves Inuyasha from Naraku.
When Sesshomaru was fighting the possessed Inuyasha (He was possessed by Magatsuhi) or whatever, the evil in the Jewel. He used Tensaiga to fend off Inuyasha instead of the Bakusaiga. When Magatsuhi mentions it, Sesshomaru says he never claimed Inuyasha as a brother but still, only used the Tenseiga.
In the movie, Sesshomaru pushed Inuyasha out of the way of a full blast from the Dragon Twister. He also could have killed Inuyasha twice during the movie but didn't.
All in all. Sesshomaru and Inuyasha will never be, come over and watch the game with me brothers. But Sesshomaru is the reason Inuyasha got stronger. In ways, Sesshomaru protected Inuyasha by teaching him to use Tetsaiga. Sesshomaru has a lot of pride, and I'm sure he's upset that Inuyasha shares the same blood as him. But throughout the course of the series I just felt that Sesshomaru had a secret desire to make Inuyasha stronger. I may be wrong, maybe Sesshomaru just wants a great battle before he kills him. But what's the purpose? Not sure. Thoughts?