Let's 1st understand what these planets represent on their own?

Sun - Sun represents Soul, Personality, King, Government, Father, Authority, Career, Top Positions, Ego, Self Esteem, Health, Education etc.

Mars - Mars represents our will power, courage, ability to take actions, aggressive nature, anger, our fighting ability, brother, male friends, a boy friend for a girl, a soldier, an athlete and real estate etc.

Rahu - It represents Illusion, Foreign Lands, Foreign Things (foreign as in unknown things. So, for someone living in Punjab, culture and rituals of Kerala is also unknown and hence Foreign), Drugs, Medicines, any type of intoxication, Fame, Wealth, Success, Obsession, Past-life incomplete Desires, Movies, Television, Online World, Cheating, Imagined Fears, Unusual Things, Unique Things, Creativity, Rule breaker. Rahu also blows things out of proportions.

3 planetary conjunction - Whenever 3 or more planets are conjunct with each other in a house/sign then things related to that house/sign become a kind of life focus for person. It is because of huge accumulation of energy within 30 degrees of sky. It means not only these 3 planets' energy is going into this house/sign but the energy of at least 4 to 6 house/sign is going into this house/sign as these planets will be ruling some houses too. So, this house/sign with conjunction becomes almost the most important house in chart. During Career Consultation, I give a huge importance to any such house which has 3 or more planets in it.

House Ruler - Besides these karaka things, Sun, Mars & Rahu will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include things related to the houses at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas.

Meaning of Sun-Mars-Rahu Conjunction - This is another stressful conjunction where all malefic planets come together. 1st of all it shows someone who is born near the solar eclipse. As such, person's relation with Father, his confidence and career can see some issues in early life and will be prospered after a certain delay. Person can have certain illusion regarding his father and career. Likewise, it also gives an illusion regarding right action of person as Mars is coming into contact with illusive Rahu. At the same time, this conjunction can make a person who is filled up with false confidence and ego or he can be just too courageous or even audacious. As Rahu is known to explode the results of the planets it is sitting with, it all depends on the dignity of Sun-Mars. If they are in good dignity then Rahu will explode those goo qualities but if they are in bad dignity, Rahu will explode the negative qualities. So, person can either be too confident or too egoist. Likewise, person can either be too courageous or too aggressive. Also, if Sun is between Mars and Rahu, then Sun will be impacted by Paap Kartari Yoga as it is only a mild-malefic planet. In such cases, things related to career, father, authority or confidence get seriously hurt. Also, things related to the house where this conjunction is placed can be achieved only after immense hard work. 

Who will win the Conjunction? - Rahu will have impact on Sun-Mars both as it will explode qualities of both planets, either good or bad. 

Dignity & Strength - It depends on dignity and strength of Sun-Mars as to how this conjunction will work. Again, Rahu will only explode what Sun-Mars provides it.