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.

Moon - Moon represents Mind, Emotions, Mother, Local Government, Peace of Mind, Home Environment, Water, Milk etc.

Saturn - It is all that which we don't like in our life. Saturn is Delay, Frustration, Restriction, Limitation and Anxiety. It is people of Authority, like Boss at your Work Place or Father-Figure at Home. Saturn is our Karmic Debt from past-life. Saturn is the things we didn't treat too well in our past life. Saturn is our Challenges and Lessons of life. Saturn is Old-Age, Teeth/Bones in Human Body, Diseases, Changes etc. Saturn represents our Step-Relations, it means that wherever Saturn sits, people related with that house give us Step Treatment. Saturn is Hard Work, Efforts and most importantly Perseverance etc. Nothing comes easy and in one go with Saturn. You have to work hard continuously over the longest possible period of life. Saturn makes a person Legend as Legends have one thing in common that they choose one field and work hard in it for whole life. 

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 in chart. 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, Moon & Saturn 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-Moon-Saturn Conjunction - So, we have another stressful conjunction here. This is a stressful conjunction for both Sun & Moon as they are conjunct with their enemy Saturn. If Moon is in between Sun & Saturn degree-wise, then it creates a kind of Paap Kartari Yoga as gentle Moon would be sitting in between 2 strong malefics. In such cases, mind will take lots of stress and can be termed as depressed mind. Even otherwise, this conjunction is going to hurt Soul (Sun) and Mind (Moon). As Saturn represents Fear, this person will be very fearful in early life. He can have stage phobia too. Close proximity of Saturn & Moon shows deep emotional/psychological issues or issues with Mother. They will have delayed beginning and success in career. Basically, this conjunction shows a repressed personality till 30 at least as Sun & Moon both are impacted by Saturn. It also shows a probable distant relation with parents. Positively, Moon-Saturn together shows a person who can meditate easily. Sun-Saturn conjunction can make person very disciplined, hard working and true to his words. In later life, they can be in an authoritative position in career but in early life, this can bring some serious issues & disagreements with authority. Again, if any of or both the planets are combust then we have to take the combust effect too. 

Who will win the Conjunction? - Saturn, on any given day. 

Dignity & Strength - 

As Saturn is involved, results will be more or less same, as discussed above, in any house/sign.